#set document(title: "5.1 Algebra with Trigonometric Ratios", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 5.1#h(0.6em)Algebra with Trigonometric Ratios In this chapter we apply some techniques from algebra to analyze more complicated trigonometric expressions. Before we begin, let's review some algebraic terminology. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Algebra Refresher] - An algebraic #strong[expression] is any meaningful collection of numbers, variables, and operation symbols. For example, the height of a golf ball is given in feet by the expression #math.equation(block: false, alt: "minus 16 t squared plus 64 t")[$− 16 t^(2) + 64 t$], where #math.equation(block: false, alt: "t")[$t$] is the number of seconds after the ball is hit. - We #strong[evaluate] an expression by substituting a specific value for the variable or variables involved. Thus, after 1 second, the height of the golf ball is #math.equation(block: true, alt: "minus 16 open parenthesis 1 close parenthesis squared plus 64 open parenthesis 11 close parenthesis equals minus 16 plus 64 equals 48 feet")[$− 16 ( 1 )^(2) + 64 ( 1 1 ) = − 16 + 64 = 48 " " "feet"$] and after 2 seconds, the height is #math.equation(block: true, alt: "minus 16 open parenthesis 2 close parenthesis squared plus 64 open parenthesis 2 close parenthesis equals minus 64 plus 128 equals 64 feet")[$− 16 ( 2 )^(2) + 64 ( 2 ) = − 64 + 128 = 64 " " "feet"$] and so on. ] === Evaluating Trigonometric Expressions Trigonometric ratios represent numbers, and they may appear as part of an algebraic expression. Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Keep in mind that #math.equation(block: false, alt: "sin ⁡ open parenthesis X close parenthesis")[$sin ( X )$], for example, is not a product: it does #emph[not] mean "sin times X" (whatever that might be). Instead, "sin" is the name of a function, and #math.equation(block: false, alt: "sin ⁡ open parenthesis X close parenthesis")[$sin ( X )$] means to evaluate the sine function at X. Thus, #math.equation(block: false, alt: "sin ⁡ open parenthesis X close parenthesis")[$sin ( X )$] is a single number, the output of the sine function. ] #examplebox("Example 1")[][ Evaluate each expression for #math.equation(block: false, alt: "X equals 30 degrees")[$X = 30^(∘)$] and #math.equation(block: false, alt: "Y equals 135 degrees")[$Y = 135^(∘)$]. + #math.equation(block: false, alt: "2 tan ⁡ open parenthesis Y close parenthesis plus 3 sin ⁡ open parenthesis X close parenthesis")[$2 tan ( Y ) + 3 sin ( X )$] + #math.equation(block: false, alt: "6 tan ⁡ open parenthesis X close parenthesis cos ⁡ open parenthesis Y close parenthesis")[$6 tan ( X ) cos ( Y )$] #solutionbox[ + Substituting the values for #math.equation(block: false, alt: "X")[$X$] and #math.equation(block: false, alt: "Y")[$Y$], we get #math.equation(block: true, alt: "2 tan ⁡ open parenthesis 135 degrees close parenthesis plus 3 sin ⁡ open parenthesis 30 degrees close parenthesis")[$2 tan ( 135^(∘) ) + 3 sin ( 30^(∘) )$] Next, we evaluate each trig ratio and follow the order of operations. #math.equation(block: true, alt: "2 tan ⁡ open parenthesis 135 degrees close parenthesis plus 3 sin ⁡ open parenthesis 30 degrees close parenthesis, equals 2 open parenthesis minus 1 close parenthesis plus 3 open parenthesis the fraction 1 over 2 close parenthesis; equals minus 2 plus the fraction 3 over 2 equals the fraction minus 1 over 2")[$2 tan ( 135^(∘) ) + 3 sin ( 30^(∘) ) & = 2 ( − 1 ) + 3 ( display(frac(1, 2)) ) \ & = − 2 + display(frac(3, 2)) = display(frac(− 1, 2))$] + This expression includes the product of two trig ratios, #math.equation(block: false, alt: "tan ⁡ open parenthesis X close parenthesis cos ⁡ open parenthesis Y close parenthesis")[$tan ( X ) cos ( Y )$]. #math.equation(block: true, alt: "6 tan ⁡ open parenthesis X close parenthesis cos ⁡ open parenthesis Y close parenthesis, equals 6 tan ⁡ open parenthesis 30 degrees close parenthesis cos ⁡ open parenthesis 135 degrees close parenthesis; equals 6 open parenthesis the fraction 1 over the square root of 3 close parenthesis open parenthesis the fraction minus 1 over the square root of 2 close parenthesis; equals the fraction minus 6 over the square root of 6 equals minus the square root of 6")[$6 tan ( X ) cos ( Y ) & = 6 tan ( 30^(∘) ) cos ( 135^(∘) ) \ & = 6 ( frac(1, sqrt(3)) ) ( display(frac(− 1, sqrt(2))) ) \ & = display(frac(− 6, sqrt(6))) = − sqrt(6)$] ] ] Evaluate each expression for #math.equation(block: false, alt: "X equals 30 degrees")[$X = 30^(∘)$], #math.equation(block: false, alt: "Y equals 60 degrees")[$Y = 60^(∘)$]. + #math.equation(block: false, alt: "4 sin ⁡ open parenthesis 3 X plus 45 degrees close parenthesis")[$4 sin ( 3 X + 45^(∘) )$] + #math.equation(block: false, alt: "1 minus cos ⁡ open parenthesis 4 Y close parenthesis")[$1 − cos ( 4 Y )$] + #math.equation(block: false, alt: "2 the square root of 2")[$2 sqrt(2)$] + #math.equation(block: false, alt: "the fraction 3 over 2")[$display(frac(3, 2))$] #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ In the previous Exercise, #math.equation(block: false, alt: "sin ⁡ open parenthesis 3 X plus 45 degrees close parenthesis")[$sin ( 3 X + 45^(∘) )$] is #emph[not] equal to #math.equation(block: false, alt: "sin ⁡ open parenthesis 3 X close parenthesis plus sin ⁡ open parenthesis 45 degrees close parenthesis")[$sin ( 3 X ) + sin ( 45^(∘) )$]. That is, #math.equation(block: true, alt: "sin ⁡ open parenthesis 90 degrees plus 45 degrees close parenthesis not equal to sin ⁡ open parenthesis 90 degrees close parenthesis plus sin ⁡ open parenthesis 45 degrees close parenthesis")[$sin ( 90^(∘) + 45^(∘) ) ≠ sin ( 90^(∘) ) + sin ( 45^(∘) )$] (You can check this for yourself.) We must follow the order of operations and evaluate the expression #math.equation(block: false, alt: "3 X plus 45 degrees")[$3 X + 45^(∘)$] inside parentheses before applying the sine function. ] === Simplifying Trigonometric Expressions Because expressions such as #math.equation(block: false, alt: "sin ⁡ open parenthesis x close parenthesis")[$sin ( x )$] and #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] are just variables, we can use algebra skills to simplify expressions involving the trig functions. For the rest of this section, we'll try to illustrate all the skills you will need going forward with your study of trigonometry. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[More Algebra] When we #strong[simplify] an algebraic expression, we obtain a new expression that has the same values as the old one, but is easier to work with. For example, we can apply the distributive law and combine like terms to simplify #math.equation(block: true, alt: "2 x open parenthesis x minus 6 close parenthesis plus 3 open parenthesis x plus 2 close parenthesis, equals 2 x squared minus 12 x plus 3 x plus 6; equals 2 x squared minus 9 x plus 6")[$2 x ( x − 6 ) + 3 ( x + 2 ) & = 2 x^(2) − 12 x + 3 x + 6 \ & = 2 x^(2) − 9 x + 6$] The new expression is #strong[equivalent] to the old one, that is, the expressions have the same value when we evaluate them at any value of #math.equation(block: false, alt: "x")[$x$]. For instance, you can check that, at #math.equation(block: false, alt: "x equals 3")[$x = 3$], #math.equation(block: true, alt: "2 open parenthesis 3 close parenthesis open parenthesis 3 minus 6 close parenthesis plus 3 open parenthesis 3 plus 2 close parenthesis, equals 6 open parenthesis minus 3 close parenthesis plus 3 open parenthesis 5 close parenthesis equals minus 3; and 2 open parenthesis 3 close parenthesis squared minus 9 open parenthesis 3 close parenthesis plus 6, equals 18 minus 27 plus 6 equals minus 3")[$2 ( 3 ) ( 3 − 6 ) + 3 ( 3 + 2 ) & = 6 ( − 3 ) + 3 ( 5 ) = − 3 \ "and" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 2 ( 3 )^(2) − 9 ( 3 ) + 6 & = 18 − 27 + 6 = − 3$] ] To simplify an expression containing trig ratios, we treat each ratio as a single variable. Compare the two calculations below: #math.equation(block: true, alt: "8 x y, minus, 6 x y, equals 2 x y; 8 cos ⁡ open parenthesis θ close parenthesis sin ⁡ open parenthesis θ close parenthesis, minus, 6 cos ⁡ open parenthesis θ close parenthesis sin ⁡ open parenthesis θ close parenthesis, equals 2 cos ⁡ open parenthesis θ close parenthesis sin ⁡ open parenthesis θ close parenthesis")[$& 8 x y & & − & & 6 x y & & = 2 x y \ & 8 cos ( θ ) sin ( θ ) & & − & & 6 cos ( θ ) sin ( θ ) & & = 2 cos ( θ ) sin ( θ )$] Both calculations are examples of combining #strong[like terms]. In the second calculation, we treat #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$] and #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] as variables, just as we treat #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] as variables in the first calculation. #examplebox("Example 2")[][ Simplify. + #math.equation(block: false, alt: "3 tan ⁡ open parenthesis A close parenthesis plus 4 tan ⁡ open parenthesis A close parenthesis minus 2 cos ⁡ open parenthesis A close parenthesis")[$3 tan ( A ) + 4 tan ( A ) − 2 cos ( A )$] + #math.equation(block: false, alt: "2 minus sin ⁡ open parenthesis B close parenthesis plus 2 sin ⁡ open parenthesis B close parenthesis")[$2 − sin ( B ) + 2 sin ( B )$] #solutionbox[ + Combine like terms. #math.equation(block: true, alt: "3 tan ⁡ open parenthesis A close parenthesis plus 4 tan ⁡ open parenthesis A close parenthesis minus 2 cos ⁡ open parenthesis A close parenthesis equals 7 tan ⁡ open parenthesis A close parenthesis minus 2 cos ⁡ open parenthesis A close parenthesis")[$3 tan ( A ) + 4 tan ( A ) − 2 cos ( A ) = 7 tan ( A ) − 2 cos ( A )$] Note that #math.equation(block: false, alt: "tan ⁡ open parenthesis A close parenthesis")[$tan ( A )$] and #math.equation(block: false, alt: "cos ⁡ open parenthesis A close parenthesis")[$cos ( A )$] are not like terms. + Combine like terms. #math.equation(block: true, alt: "2 minus sin ⁡ open parenthesis B close parenthesis plus 2 sin ⁡ open parenthesis B close parenthesis equals 2 plus sin ⁡ open parenthesis B close parenthesis")[$2 − sin ( B ) + 2 sin ( B ) = 2 + sin ( B )$] Note that #math.equation(block: false, alt: "minus sin ⁡ open parenthesis B close parenthesis")[$" " − sin ( B )$] is treated as #math.equation(block: false, alt: "minus 1 times sin ⁡ open parenthesis B close parenthesis")[$" " − 1 ⋅ sin ( B )$]. ] ] Simplify #math.equation(block: true, alt: "2 cos ⁡ open parenthesis t close parenthesis minus 4 cos ⁡ open parenthesis w close parenthesis sin ⁡ open parenthesis w close parenthesis plus 3 cos ⁡ open parenthesis t close parenthesis minus 2 cos ⁡ open parenthesis w close parenthesis")[$" " 2 cos ( t ) − 4 cos ( w ) sin ( w ) + 3 cos ( t ) − 2 cos ( w )$] #math.equation(block: true, alt: "5 cos ⁡ open parenthesis t close parenthesis minus 4 cos ⁡ open parenthesis w close parenthesis sin ⁡ open parenthesis w close parenthesis minus 2 cos ⁡ open parenthesis w close parenthesis")[$5 cos ( t ) − 4 cos ( w ) sin ( w ) − 2 cos ( w )$] #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ In the previous Checkpoint, note that #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis")[$cos ( t )$] and #math.equation(block: false, alt: "cos ⁡ open parenthesis w close parenthesis")[$cos ( w )$] are not like terms. (We can choose values for #math.equation(block: false, alt: "t")[$t$] and #math.equation(block: false, alt: "w")[$w$] so that #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis")[$cos ( t )$] and #math.equation(block: false, alt: "cos ⁡ open parenthesis w close parenthesis")[$cos ( w )$] have different values.) ] #examplebox("Example 3")[][ Simplify, and evaluate for #math.equation(block: false, alt: "z equals 40 degrees")[$z = 40^(∘)$]. #math.equation(block: true, alt: "3 sin ⁡ open parenthesis z close parenthesis minus sin ⁡ open parenthesis z close parenthesis tan ⁡ open parenthesis z close parenthesis plus 3 sin ⁡ open parenthesis z close parenthesis")[$3 sin ( z ) − sin ( z ) tan ( z ) + 3 sin ( z )$] #solutionbox[ We can combine like terms to get #math.equation(block: true, alt: "6 sin ⁡ open parenthesis z close parenthesis minus sin ⁡ open parenthesis z close parenthesis tan ⁡ open parenthesis z close parenthesis")[$6 sin ( z ) − sin ( z ) tan ( z )$] Because #math.equation(block: false, alt: "40 degrees")[$40^(∘)$] is not one of the angles for which we know exact trig values, we use a calculator to evaluate the expression. #math.equation(block: true, alt: "6 sin ⁡ open parenthesis 40 degrees close parenthesis minus sin ⁡ open parenthesis 40 degrees close parenthesis tan ⁡ open parenthesis 40 degrees close parenthesis, equals 6 open parenthesis 0.6428 close parenthesis minus open parenthesis 0.6428 close parenthesis open parenthesis 0.8391 close parenthesis; equals 3.3174")[$6 sin ( 40^(∘) ) − sin ( 40^(∘) ) tan ( 40^(∘) ) & = 6 ( 0.6428 ) − ( 0.6428 ) ( 0.8391 ) \ & = 3.3174$] ] ] Simplify, and evaluate for #math.equation(block: false, alt: "x equals 25 degrees , y equals 70 degrees")[$x = 25^(∘) , " " y = 70^(∘)$] #math.equation(block: true, alt: "3 cos ⁡ open parenthesis x close parenthesis plus cos ⁡ open parenthesis y close parenthesis minus 2 cos ⁡ open parenthesis y close parenthesis plus cos ⁡ open parenthesis x close parenthesis")[$3 cos ( x ) + cos ( y ) − 2 cos ( y ) + cos ( x )$] #math.equation(block: true, alt: "3.2832")[$3.2832$] === Powers of Trigonometric Ratios Compare the two expressions #math.equation(block: true, alt: "open parenthesis cos ⁡ open parenthesis θ close parenthesis close parenthesis squared and cos ⁡ open parenthesis θ squared close parenthesis")[$( cos ( θ ) )^(2) " " " " "and" " " " " " " cos ( θ^(2) )$] They are not the same. - The first expression, #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$( cos ( θ ) )^(2)$], says to compute #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$] and then square the result. - #math.equation(block: false, alt: "cos ⁡ open parenthesis θ squared close parenthesis")[$cos ( θ^(2) )$] says to square the angle first, and then compute the cosine. For example, if #math.equation(block: false, alt: "θ equals 30 degrees")[$θ = 30^(∘)$], then #math.equation(block: true, alt: "open parenthesis cos ⁡ open parenthesis 30 degrees close parenthesis close parenthesis squared, equals open parenthesis the fraction the square root of 3 over 2 close parenthesis squared equals the fraction 3 over 4; but cos ⁡ open parenthesis open parenthesis 30 squared close parenthesis degrees close parenthesis, equals cos ⁡ open parenthesis 900 degrees close parenthesis equals cos ⁡ open parenthesis 180 degrees close parenthesis equals minus 1")[$( cos ( 30^(∘) ) )^(2) & = attach(( display(frac(sqrt(3), 2)) ), t: 2) = display(frac(3, 4)) \ "but" " " " " " " " " " " " " " " " " cos ( ( 30^(2) )^(∘) ) & = cos ( 900^(∘) ) = cos ( 180^(∘) ) = − 1$] We usually write #math.equation(block: false, alt: "cos squared ⁡ open parenthesis θ close parenthesis")[$cos^(2) ( θ )$] instead of #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$attach(( cos ( θ ) ), t: 2)$], to distinguish it from #math.equation(block: false, alt: "cos ⁡ open parenthesis θ squared close parenthesis")[$cos ( θ^(2) )$], and to reduce the number of parentheses. Thus, #math.equation(block: false, alt: "cos squared ⁡ open parenthesis θ close parenthesis")[$cos^(2) ( θ )$] means the square of #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The square of cosine] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis θ close parenthesis means open parenthesis cos ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$cos^(2) ( θ ) " " " " " " " " "means" " " " " " " " " " " ( cos ( θ ) )^(2)$] ] The same notation applies to the other trig ratios, so that #math.equation(block: true, alt: "sin squared ⁡ open parenthesis θ close parenthesis equals open parenthesis sin ⁡ open parenthesis θ close parenthesis close parenthesis squared and tan squared ⁡ open parenthesis θ close parenthesis equals open parenthesis tan ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$sin^(2) ( θ ) = ( sin ( θ ) )^(2) " " " " " " "and" " " " " " " tan^(2) ( θ ) = ( tan ( θ ) )^(2)$] #examplebox("Example 4")[][ Evaluate #math.equation(block: false, alt: "sin squared ⁡ open parenthesis 45 degrees close parenthesis")[$sin^(2) ( 45^(∘) )$]. #solutionbox[ #math.equation(block: true, alt: "sin squared ⁡ open parenthesis 45 degrees close parenthesis equals open parenthesis sin ⁡ open parenthesis 45 degrees close parenthesis close parenthesis squared equals open parenthesis the fraction 1 over the square root of 2 close parenthesis squared equals the fraction 1 over 2")[$sin^(2) ( 45^(∘) ) = ( sin ( 45^(∘) ) )^(2) = attach(( display(frac(1, sqrt(2))) ), t: 2) = display(frac(1, 2))$] ] ] Other powers are written in the same fashion. Thus, for example, #math.equation(block: false, alt: "sin cubed ⁡ open parenthesis θ close parenthesis equals open parenthesis sin ⁡ open parenthesis θ close parenthesis close parenthesis cubed")[$sin^(3) ( θ ) = ( sin ( θ ) )^(3)$]. Evaluate #math.equation(block: false, alt: "tan to the power 4 ⁡ open parenthesis 60 degrees close parenthesis")[$tan^(4) ( 60^(∘) )$] #math.equation(block: true, alt: "9")[$9$] === Products We can multiply together trigonometric expressions, just as we multiply algebraic expressions. Recall that we use the #strong[distributive law] in computing products such as #math.equation(block: true, alt: "x open parenthesis 3 x minus 2 close parenthesis equals 3 x squared minus 2 x")[$x ( 3 x − 2 ) = 3 x^(2) − 2 x$] and #math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 5 close parenthesis equals x squared plus 2 x minus 15")[$( x − 3 ) ( x + 5 ) = x^(2) + 2 x − 15$] #examplebox("Example 5")[][ Using the distributive law, multiply #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis open parenthesis 3 cos ⁡ open parenthesis t close parenthesis minus 2 close parenthesis")[$" " " " cos ( t ) ( 3 cos ( t ) − 2 )$]. #solutionbox[ Think of #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis")[$cos ( t )$] as a single variable, and multiply by each term inside parentheses. (The algebraic form of the calculation is shown below in blue). #math.equation(block: true, alt: "cos ⁡ open parenthesis t close parenthesis open parenthesis 3 cos ⁡ open parenthesis t close parenthesis minus 2 close parenthesis, equals open parenthesis cos ⁡ open parenthesis t close parenthesis close parenthesis open parenthesis 3 cos ⁡ open parenthesis t close parenthesis close parenthesis minus open parenthesis cos ⁡ open parenthesis t close parenthesis close parenthesis times 2; equals 3 cos squared ⁡ open parenthesis t close parenthesis minus 2 cos ⁡ open parenthesis t close parenthesis; x open parenthesis 3 x minus 2 close parenthesis, equals x times 3 x minus x times 2; equals 3 x squared minus 2 x")[$cos ( t ) ( 3 cos ( t ) − 2 ) & = ( cos ( t ) ) ( 3 cos ( t ) ) − ( cos ( t ) ) ⋅ 2 \ & = 3 cos^(2) ( t ) − 2 cos ( t ) \ bold(italic(x)) ( 3 bold(italic(x)) − 2 ) & = bold(italic(x)) ⋅ 3 bold(italic(x)) − bold(italic(x)) ⋅ 2 \ & = 3 bold(italic(x))^(2) − 2 bold(italic(x))$] Notice that we write #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis t close parenthesis close parenthesis squared")[$( cos ( t ) )^(2)$] as #math.equation(block: false, alt: "cos squared ⁡ open parenthesis t close parenthesis")[$cos^(2) ( t )$]. ] ] Multiply #math.equation(block: false, alt: "2 tan ⁡ open parenthesis β close parenthesis open parenthesis 4 tan squared ⁡ open parenthesis β close parenthesis plus tan ⁡ open parenthesis α close parenthesis close parenthesis")[$" " " " 2 tan ( β ) ( 4 tan^(2) ( β ) + tan ( α ) )$] #math.equation(block: true, alt: "8 tan cubed ⁡ open parenthesis β close parenthesis plus 2 tan ⁡ open parenthesis β close parenthesis tan ⁡ open parenthesis α close parenthesis")[$8 tan^(3) ( β ) + 2 tan ( β ) tan ( α )$] We can also use the distributive law to multiply binomials that include trig ratios. You may have used the acronym #math.equation(block: false, alt: "F O I L")[$bold(italic(F)) bold(italic(O)) bold(italic(I)) bold(italic(L))$] to remember the four multiplications in a product of binomials: #math.equation(block: false, alt: "F")[$" " " " " " " " " " " " " " " " bold(italic(F))$]irst terms, #math.equation(block: false, alt: "O")[$bold(italic(O))$]utside terms, #math.equation(block: false, alt: "I")[$bold(italic(I))$]nside terms, and #math.equation(block: false, alt: "L")[$bold(italic(L))$]ast terms. #examplebox("Example 6")[][ Multiply #math.equation(block: false, alt: "open parenthesis 4 sin ⁡ open parenthesis C close parenthesis minus 1 close parenthesis open parenthesis 3 sin ⁡ open parenthesis C close parenthesis plus 2 close parenthesis")[$" " " " ( 4 sin ( C ) − 1 ) ( 3 sin ( C ) + 2 )$]. #solutionbox[ This calculation is similar to the product #math.equation(block: false, alt: "open parenthesis 4 x minus 1 close parenthesis open parenthesis 3 x plus 2 close parenthesis")[$( 4 x − 1 ) ( 3 x + 2 )$], except that the variable #math.equation(block: false, alt: "x")[$x$] has been replaced by #math.equation(block: false, alt: "sin ⁡ open parenthesis C close parenthesis")[$sin ( C )$]. Compare the calculations for the two products; first the familiar algebraic product: #math.equation(block: true, alt: "open parenthesis 4 x minus 1 close parenthesis open parenthesis 3 x plus 2 close parenthesis, equals 4 x times 3 x plus 4 x times 2 minus 1 times 3 x minus 1 times 2; equals 12 x squared plus 8 x minus 3 x minus 2; equals 12 x squared plus 5 x minus 2")[$( 4 x − 1 ) ( 3 x + 2 ) & = 4 x ⋅ 3 x + 4 x ⋅ 2 − 1 ⋅ 3 x − 1 ⋅ 2 \ & = 12 x^(2) + 8 x − 3 x − 2 \ & = 12 x^(2) + 5 x − 2$] We compute the product in this example in the same way, but replacing #math.equation(block: false, alt: "x")[$x$] by #math.equation(block: false, alt: "sin ⁡ open parenthesis C close parenthesis")[$sin ( C )$]. #math.equation(block: true, alt: "open parenthesis 4, sin ⁡ open parenthesis C close parenthesis minus 1 close parenthesis open parenthesis sin ⁡ open parenthesis C close parenthesis plus 2 close parenthesis; equals open parenthesis 4 sin ⁡ open parenthesis C close parenthesis close parenthesis open parenthesis 3 sin ⁡ open parenthesis C close parenthesis close parenthesis plus open parenthesis 4 sin ⁡ open parenthesis C close parenthesis close parenthesis times 2 minus 1 open parenthesis 3 sin ⁡ open parenthesis C close parenthesis close parenthesis minus 1 times 2; equals 12 sin squared ⁡ open parenthesis C close parenthesis plus 5 sin ⁡ open parenthesis C close parenthesis minus 2")[$( 4 & sin ( C ) − 1 ) ( sin ( C ) + 2 ) \ & = ( 4 sin ( C ) ) ( 3 sin ( C ) ) + ( 4 sin ( C ) ) ⋅ 2 − 1 ( 3 sin ( C ) ) − 1 ⋅ 2 \ & = 12 sin^(2) ( C ) + 5 sin ( C ) − 2$] ] ] Expand #math.equation(block: false, alt: "open parenthesis 4 cos ⁡ open parenthesis α close parenthesis plus 3 close parenthesis squared")[$" " " " ( 4 cos ( α ) + 3 )^(2)$] #math.equation(block: true, alt: "16 cos squared ⁡ open parenthesis α close parenthesis plus 24 cos ⁡ open parenthesis α close parenthesis plus 9")[$16 cos^(2) ( α ) + 24 cos ( α ) + 9$] === Factoring We can factor trigonometric expressions with the same techniques we use for algebraic expressions. In the next two Examples, compare the familiar algebraic factoring with a similar trigonometric expression. #examplebox("Example 7")[][ Factor. + #math.equation(block: false, alt: "6 w squared minus 9 w")[$6 w^(2) − 9 w$] + #math.equation(block: false, alt: "6 sin squared ⁡ open parenthesis θ close parenthesis minus 9 sin ⁡ open parenthesis θ close parenthesis")[$6 sin^(2) ( θ ) − 9 sin ( θ )$] #solutionbox[ + We factor out the common factor, #math.equation(block: false, alt: "3 w")[$3 w$]. #math.equation(block: true, alt: "6 w squared minus 9 w equals 3 w open parenthesis 2 w minus 3 close parenthesis")[$6 w^(2) − 9 w = 3 w ( 2 w − 3 )$] + We factor out the common factor, #math.equation(block: false, alt: "3 sin ⁡ open parenthesis θ close parenthesis")[$3 sin ( θ )$]. #math.equation(block: true, alt: "6 sin squared ⁡ open parenthesis θ close parenthesis minus 9 sin ⁡ open parenthesis θ close parenthesis equals 3 sin ⁡ open parenthesis θ close parenthesis open parenthesis 2 sin ⁡ open parenthesis θ close parenthesis minus 3 close parenthesis")[$6 sin^(2) ( θ ) − 9 sin ( θ ) = 3 sin ( θ ) ( 2 sin ( θ ) − 3 )$] ] ] Factor. + #math.equation(block: false, alt: "2 a squared minus a b")[$2 a^(2) − a b$] + #math.equation(block: false, alt: "2 cos squared ⁡ open parenthesis ϕ close parenthesis minus cos ⁡ open parenthesis ϕ close parenthesis sin ⁡ open parenthesis ϕ close parenthesis")[$2 cos^(2) ( ϕ ) − cos ( ϕ ) sin ( ϕ )$] + #math.equation(block: false, alt: "a open parenthesis 2 a minus b close parenthesis")[$a ( 2 a − b )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis ϕ close parenthesis open parenthesis 2 cos ⁡ open parenthesis ϕ close parenthesis minus sin ⁡ open parenthesis ϕ close parenthesis close parenthesis")[$cos ( ϕ ) ( 2 cos ( ϕ ) − sin ( ϕ ) )$] We can also factor quadratic trinomials. #examplebox("Example 8")[][ Factor. + #math.equation(block: false, alt: "t squared minus 3 t minus 10")[$t^(2) − 3 t − 10$] + #math.equation(block: false, alt: "tan squared ⁡ open parenthesis α close parenthesis minus 3 tan ⁡ open parenthesis α close parenthesis minus 10")[$tan^(2) ( α ) − 3 tan ( α ) − 10$] #solutionbox[ + We look for numbers #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$] so that #math.equation(block: false, alt: "open parenthesis t plus p close parenthesis open parenthesis t plus q close parenthesis equals t squared minus 3 t minus 10")[$( t + p ) ( t + q ) = t^(2) − 3 t − 10$]. We see that their product must be #math.equation(block: false, alt: "p q equals minus 10")[$" " p q = − 10 " "$], and their sum must be #math.equation(block: false, alt: "p plus q equals minus 3")[$" " p + q = − 3$]. By checking the factors of #math.equation(block: false, alt: "minus 10")[$− 10$] for the correct sum, we find #math.equation(block: false, alt: "p equals minus 5")[$p = − 5$] and #math.equation(block: false, alt: "q equals 2")[$q = 2$]. Thus, #math.equation(block: true, alt: "t squared minus 3 t minus 10 equals open parenthesis t minus 5 close parenthesis open parenthesis t plus 2 close parenthesis")[$t^(2) − 3 t − 10 = ( t − 5 ) ( t + 2 )$] + Now replace #math.equation(block: false, alt: "t")[$t$] by #math.equation(block: false, alt: "tan ⁡ open parenthesis α close parenthesis")[$tan ( α )$] to find #math.equation(block: true, alt: "tan squared ⁡ open parenthesis α close parenthesis minus 3 tan ⁡ open parenthesis α close parenthesis minus 10 equals open parenthesis tan ⁡ open parenthesis α close parenthesis minus 5 close parenthesis open parenthesis tan ⁡ open parenthesis α close parenthesis plus 2 close parenthesis")[$tan^(2) ( α ) − 3 tan ( α ) − 10 = ( tan ( α ) − 5 ) ( tan ( α ) + 2 )$] ] ] Factor. + #math.equation(block: false, alt: "3 z squared minus 2 z minus 1")[$3 z^(2) − 2 z − 1$] + #math.equation(block: false, alt: "3 sin squared ⁡ open parenthesis β close parenthesis minus 2 sin ⁡ open parenthesis β close parenthesis minus 1")[$3 sin^(2) ( β ) − 2 sin ( β ) − 1$] + #math.equation(block: false, alt: "open parenthesis 3 z plus 1 close parenthesis open parenthesis z minus 1 close parenthesis")[$( 3 z + 1 ) ( z − 1 )$] + #math.equation(block: false, alt: "open parenthesis 3 sin ⁡ open parenthesis β close parenthesis plus 1 close parenthesis open parenthesis sin ⁡ open parenthesis β close parenthesis minus 1 close parenthesis")[$( 3 sin ( β ) + 1 ) ( sin ( β ) − 1 )$] Review the following skills you will need for this section. #notebox("Algebra Refresher", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Factor. + #math.equation(block: false, alt: "2 x squared plus 5 x minus 12")[$2 x^(2) + 5 x − 12$] + #math.equation(block: false, alt: "3 x squared minus 2 x minus 8")[$3 x^(2) − 2 x − 8$] + #math.equation(block: false, alt: "12 x squared plus x minus 1")[$12 x^(2) + x − 1$] + #math.equation(block: false, alt: "6 x squared minus 13 x minus 15")[$6 x^(2) − 13 x − 15$] + #math.equation(block: false, alt: "6 x squared minus x minus 12")[$6 x^(2) − x − 12$] + #math.equation(block: false, alt: "24 x squared plus 10 x minus 21")[$24 x^(2) + 10 x − 21$] #math.equation(block: true, alt: "bar")[$underline(#h(2em) #h(2em) #h(2em) #h(2em))$] Algebra Refresher Answers + #math.equation(block: false, alt: "open parenthesis 2 x minus 3 close parenthesis open parenthesis x plus 4 close parenthesis")[$( 2 x − 3 ) ( x + 4 )$] + #math.equation(block: false, alt: "open parenthesis 3 x plus 4 close parenthesis open parenthesis x minus 2 close parenthesis")[$( 3 x + 4 ) ( x − 2 )$] + #math.equation(block: false, alt: "open parenthesis 4 x minus 1 close parenthesis open parenthesis 3 x plus 1 close parenthesis")[$( 4 x − 1 ) ( 3 x + 1 )$] + #math.equation(block: false, alt: "open parenthesis 6 x plus 5 close parenthesis open parenthesis x minus 3 close parenthesis")[$( 6 x + 5 ) ( x − 3 )$] + #math.equation(block: false, alt: "open parenthesis 2 x minus 3 close parenthesis open parenthesis 3 x plus 4 close parenthesis")[$( 2 x − 3 ) ( 3 x + 4 )$] + #math.equation(block: false, alt: "open parenthesis 6 x plus 7 close parenthesis open parenthesis 4 x minus 3 close parenthesis")[$( 6 x + 7 ) ( 4 x − 3 )$] ] === Section 5.1 Summary ==== Vocabulary - Expression - Evaluate - Binomial - Trinomial - Simplify - Equivalent expression - Like terms - Distributive law - Factor ==== Concepts + Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions. To simplify an expression containing trig ratios, we treat each ratio as a single variable. + #math.equation(block: false, alt: "sin ⁡ open parenthesis X plus Y close parenthesis")[$sin ( X + Y )$] is not equal to #math.equation(block: false, alt: "sin ⁡ open parenthesis X close parenthesis plus sin ⁡ open parenthesis Y close parenthesis")[$sin ( X ) + sin ( Y )$] (and the same holds for the other trig ratios). Remember that the parentheses indicate function notation, not multiplication. + We write #math.equation(block: false, alt: "cos squared ⁡ open parenthesis θ close parenthesis")[$cos^(2) ( θ )$] to denote #math.equation(block: false, alt: "open parenthesis cos ⁡ 9 close parenthesis squared")[$( cos 9 )^(2)$], and #math.equation(block: false, alt: "cos to the power n ⁡ open parenthesis θ close parenthesis")[$cos^(n) ( θ )$] to denote #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis θ close parenthesis close parenthesis to the power n")[$( cos ( θ ) )^(n)$]. (Similarly for the other trig ratios.) + We can factor trigonometric expressions with the same techniques we use for algebraic expressions. ==== Study Questions + To evaluate #math.equation(block: false, alt: "cos squared ⁡ open parenthesis 30 degrees close parenthesis")[$cos^(2) ( 30^(∘) )$], Delbert used the keystrokes #math.equation(block: true, alt: "COS 30 x squared ENTER")[$#box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display("COS")$] " " " " 30 " " " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(x^(2))$] " " " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display("ENTER")$]$] and got the answer 1. Were his keystrokes correct? Why or why not? + Make up an example to show that #math.equation(block: false, alt: "tan ⁡ open parenthesis θ plus ϕ close parenthesis not equal to tan ⁡ open parenthesis θ close parenthesis plus tan ⁡ open parenthesis ϕ close parenthesis")[$tan ( θ + ϕ ) ≠ tan ( θ ) + tan ( ϕ )$]. + Factor each expression, if possible.+ #math.equation(block: false, alt: "x squared minus 4 x")[$x^(2) − 4 x$] + #math.equation(block: false, alt: "x squared minus 4")[$x^(2) − 4$] + #math.equation(block: false, alt: "x squared plus 4")[$x^(2) + 4$] + #math.equation(block: false, alt: "x squared plus 4 x plus 4")[$x^(2) + 4 x + 4$] + #math.equation(block: false, alt: "x squared minus 4 x plus 4")[$x^(2) − 4 x + 4$] + #math.equation(block: false, alt: "x squared plus 4 x")[$x^(2) + 4 x$] + #math.equation(block: false, alt: "x squared minus 4 x minus 4")[$x^(2) − 4 x − 4$] + #math.equation(block: false, alt: "minus x squared plus 4")[$− x^(2) + 4$] ==== Skills + Evaluate trigonometric expressions \#1–22 + Simplify trigonometric expressions \#23–34 + Recognize equivalent expressions \#35–44 + Multiply or expand trigonometric expressions \#45–56 + Factor trigonometric expressions \#57–70 === Homework 5.1 For Problems 1–8, evaluate the expressions, using exact values for the trigonometric ratios. #math.equation(block: true, alt: "5 tan ⁡ open parenthesis 135 degrees close parenthesis plus 6 cos ⁡ open parenthesis 60 degrees close parenthesis")[$5 tan ( 135^(∘) ) + 6 cos ( 60^(∘) )$] #math.equation(block: true, alt: "minus 2")[$− 2$] #math.equation(block: true, alt: "3 tan ⁡ open parenthesis 240 degrees close parenthesis plus 8 sin ⁡ open parenthesis 300 degrees close parenthesis")[$3 tan ( 240^(∘) ) + 8 sin ( 300^(∘) )$] #math.equation(block: true, alt: "sin ⁡ open parenthesis 15 degrees plus 30 degrees close parenthesis")[$sin ( 15^(∘) + 30^(∘) )$] #math.equation(block: true, alt: "the fraction 1 over the square root of 2")[$display(frac(1, sqrt(2)))$] #math.equation(block: true, alt: "cos ⁡ open parenthesis 2 times 75 degrees close parenthesis")[$cos ( 2 ⋅ 75^(∘) )$] #math.equation(block: true, alt: "8 cos squared ⁡ open parenthesis 30 degrees close parenthesis")[$8 cos^(2) ( 30^(∘) )$] #math.equation(block: true, alt: "6")[$6$] #math.equation(block: true, alt: "12 sin squared ⁡ open parenthesis 315 degrees close parenthesis")[$12 sin^(2) ( 315^(∘) )$] #math.equation(block: true, alt: "3 tan squared ⁡ open parenthesis 150 degrees close parenthesis minus sin squared ⁡ open parenthesis 45 degrees close parenthesis")[$3 tan^(2) ( 150^(∘) ) − sin^(2) ( 45^(∘) )$] #math.equation(block: true, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] #math.equation(block: true, alt: "1 plus tan squared ⁡ open parenthesis 120 degrees close parenthesis")[$1 + tan^(2) ( 120^(∘) )$] For Problems 9–16, evaluate the expressions for #math.equation(block: false, alt: "x equals 30 degrees , y equals 45 degrees")[$x = 30^(∘) , " " y = 45^(∘)$], and #math.equation(block: false, alt: "z equals 60 degrees")[$z = 60^(∘)$]. Give exact values for your answers. #math.equation(block: true, alt: "3 sin ⁡ open parenthesis x close parenthesis plus 5 cos ⁡ open parenthesis y close parenthesis")[$3 sin ( x ) + 5 cos ( y )$] #math.equation(block: true, alt: "4")[$4$] #math.equation(block: true, alt: "4 tan ⁡ open parenthesis y close parenthesis plus 6 cos ⁡ open parenthesis y close parenthesis")[$4 tan ( y ) + 6 cos ( y )$] #math.equation(block: true, alt: "minus 2 tan ⁡ open parenthesis 3 y close parenthesis")[$− 2 tan ( 3 y )$] #math.equation(block: true, alt: "2")[$2$] #math.equation(block: true, alt: "sin ⁡ open parenthesis 3 z minus 2 x close parenthesis")[$sin ( 3 z − 2 x )$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis x close parenthesis plus sin squared ⁡ open parenthesis x close parenthesis")[$cos^(2) ( x ) + sin^(2) ( x )$] #math.equation(block: true, alt: "1")[$1$] #math.equation(block: true, alt: "7 sin squared ⁡ open parenthesis y close parenthesis plus 7 cos squared ⁡ open parenthesis y close parenthesis")[$7 sin^(2) ( y ) + 7 cos^(2) ( y )$] #math.equation(block: true, alt: "cos ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis z close parenthesis minus sin ⁡ open parenthesis x close parenthesis sin ⁡ open parenthesis z close parenthesis")[$cos ( x ) cos ( z ) − sin ( x ) sin ( z )$] #math.equation(block: true, alt: "0")[$0$] #math.equation(block: true, alt: "tan ⁡ open parenthesis 180 degrees minus x close parenthesis tan ⁡ open parenthesis x close parenthesis")[$tan ( 180^(∘) − x ) tan ( x )$] For Problems 17–22,evaluate the expressions using a calculator. + #math.equation(block: false, alt: "sin ⁡ open parenthesis 10 degrees plus 40 degrees close parenthesis")[$sin ( 10^(∘) + 40^(∘) )$] + #math.equation(block: false, alt: "sin ⁡ open parenthesis 10 degrees close parenthesis plus sin ⁡ open parenthesis 40 degrees close parenthesis")[$sin ( 10^(∘) ) + sin ( 40^(∘) )$] + #math.equation(block: false, alt: "sin ⁡ open parenthesis 10 degrees close parenthesis cos ⁡ open parenthesis 40 degrees close parenthesis plus cos ⁡ open parenthesis 10 degrees close parenthesis sin ⁡ open parenthesis 40 degrees close parenthesis")[$sin ( 10^(∘) ) cos ( 40^(∘) ) + cos ( 10^(∘) ) sin ( 40^(∘) )$] + #math.equation(block: false, alt: "0.7660")[$0.7660$] + #math.equation(block: false, alt: "0.8164")[$0.8164$] + #math.equation(block: false, alt: "0.7660")[$0.7660$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 20 degrees plus 50 degrees close parenthesis")[$cos ( 20^(∘) + 50^(∘) )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 20 degrees close parenthesis plus cos ⁡ open parenthesis 50 degrees close parenthesis")[$cos ( 20^(∘) ) + cos ( 50^(∘) )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 20 degrees close parenthesis cos ⁡ open parenthesis 50 degrees close parenthesis minus sin ⁡ open parenthesis 20 degrees close parenthesis sin ⁡ open parenthesis 50 degrees close parenthesis")[$cos ( 20^(∘) ) cos ( 50^(∘) ) − sin ( 20^(∘) ) sin ( 50^(∘) )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 2 times 24 degrees close parenthesis")[$cos ( 2 ⋅ 24^(∘) )$] + #math.equation(block: false, alt: "2 cos ⁡ open parenthesis 24 degrees close parenthesis")[$2 cos ( 24^(∘) )$] + #math.equation(block: false, alt: "2 cos squared ⁡ open parenthesis 24 degrees close parenthesis minus 1")[$2 cos^(2) ( 24^(∘) ) − 1$] + #math.equation(block: false, alt: "0.6691")[$0.6691$] + #math.equation(block: false, alt: "1.8271")[$1.8271$] + #math.equation(block: false, alt: "0.6691")[$0.6691$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 3 times 49 degrees close parenthesis")[$cos ( 3 ⋅ 49^(∘) )$] + #math.equation(block: false, alt: "3 cos ⁡ open parenthesis 49 degrees close parenthesis")[$3 cos ( 49^(∘) )$] + #math.equation(block: false, alt: "4 cos cubed ⁡ open parenthesis 49 degrees close parenthesis minus 3 cos ⁡ open parenthesis 49 degrees close parenthesis")[$4 cos^(3) ( 49^(∘) ) − 3 cos ( 49^(∘) )$] + #math.equation(block: false, alt: "cos squared ⁡ open parenthesis 17 degrees close parenthesis plus sin squared ⁡ open parenthesis 17 degrees close parenthesis")[$cos^(2) ( 17^(∘) ) + sin^(2) ( 17^(∘) )$] + #math.equation(block: false, alt: "cos squared ⁡ open parenthesis 86 degrees close parenthesis plus sin squared ⁡ open parenthesis 86 degrees close parenthesis")[$cos^(2) ( 86^(∘) ) + sin^(2) ( 86^(∘) )$] + #math.equation(block: false, alt: "cos squared ⁡ open parenthesis 111 degrees close parenthesis plus sin squared ⁡ open parenthesis 111 degrees close parenthesis")[$cos^(2) ( 111^(∘) ) + sin^(2) ( 111^(∘) )$] + #math.equation(block: false, alt: "1")[$1$] + #math.equation(block: false, alt: "1")[$1$] + #math.equation(block: false, alt: "1")[$1$] + #math.equation(block: false, alt: "the fraction 1 over cos squared ⁡ open parenthesis 25 degrees close parenthesis minus tan squared ⁡ open parenthesis 25 degrees close parenthesis")[$display(frac(1, cos^(2) ( 25^(∘) ))) − tan^(2) ( 25^(∘) )$] + #math.equation(block: false, alt: "the fraction 1 over cos squared ⁡ open parenthesis 100 degrees close parenthesis minus tan squared ⁡ open parenthesis 100 degrees close parenthesis")[$display(frac(1, cos^(2) ( 100^(∘) ))) − tan^(2) ( 100^(∘) )$] + #math.equation(block: false, alt: "the fraction 1 over cos squared ⁡ open parenthesis 8 degrees close parenthesis minus tan squared ⁡ open parenthesis 8 degrees close parenthesis")[$display(frac(1, cos^(2) ( 8^(∘) ))) − tan^(2) ( 8^(∘) )$] For Problems 23–28, combine like terms. + #math.equation(block: false, alt: "3 x squared minus x minus 5 x squared")[$3 x^(2) − x − 5 x^(2)$] + #math.equation(block: false, alt: "3 cos squared ⁡ open parenthesis θ close parenthesis minus cos ⁡ open parenthesis θ close parenthesis minus 5 cos squared ⁡ open parenthesis θ close parenthesis")[$3 cos^(2) ( θ ) − cos ( θ ) − 5 cos^(2) ( θ )$] + #math.equation(block: false, alt: "minus 2 x squared minus x")[$− 2 x^(2) − x$] + #math.equation(block: false, alt: "minus 2 cos squared ⁡ open parenthesis θ close parenthesis minus cos ⁡ open parenthesis θ close parenthesis")[$− 2 cos^(2) ( θ ) − cos ( θ )$] + #math.equation(block: false, alt: "4 x plus 5 y minus y")[$4 x + 5 y − y$] + #math.equation(block: false, alt: "4 cos ⁡ open parenthesis θ close parenthesis plus 5 sin ⁡ open parenthesis θ close parenthesis minus sin ⁡ open parenthesis θ close parenthesis")[$4 cos ( θ ) + 5 sin ( θ ) − sin ( θ )$] + #math.equation(block: false, alt: "minus 3 S C plus 7 S C")[$− 3 S C + 7 S C$] + #math.equation(block: false, alt: "minus 3 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis plus 7 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis")[$− 3 sin ( θ ) cos ( θ ) + 7 sin ( θ ) cos ( θ )$] + #math.equation(block: false, alt: "4 S C")[$4 S C$] + #math.equation(block: false, alt: "4 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis")[$4 sin ( θ ) cos ( θ )$] + #math.equation(block: false, alt: "4 S C plus 11 S C minus 17 S C")[$4 S C + 11 S C − 17 S C$] + #math.equation(block: false, alt: "4 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis plus 11 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis minus 17 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis")[$4 sin ( θ ) cos ( θ ) + 11 sin ( θ ) cos ( θ ) − 17 sin ( θ ) cos ( θ )$] + #math.equation(block: false, alt: "minus C squared S cubed plus 6 C squared S cubed")[$− C^(2) S^(3) + 6 C^(2) S^(3)$] + #math.equation(block: false, alt: "minus cos squared ⁡ open parenthesis θ close parenthesis sin cubed ⁡ open parenthesis θ close parenthesis plus 6 cos squared ⁡ open parenthesis θ close parenthesis sin cubed ⁡ open parenthesis θ close parenthesis")[$− cos^(2) ( θ ) sin^(3) ( θ ) + 6 cos^(2) ( θ ) sin^(3) ( θ )$] + #math.equation(block: false, alt: "5 C squared S cubed")[$5 C^(2) S^(3)$] + #math.equation(block: false, alt: "5 cos squared ⁡ open parenthesis θ close parenthesis sin cubed ⁡ open parenthesis θ close parenthesis")[$5 cos^(2) ( θ ) sin^(3) ( θ )$] + #math.equation(block: false, alt: "7 C squared S squared minus open parenthesis 2 C S close parenthesis squared")[$7 C^(2) S^(2) − ( 2 C S )^(2)$] + #math.equation(block: false, alt: "7 cos squared ⁡ open parenthesis θ close parenthesis sin squared ⁡ open parenthesis θ close parenthesis minus open parenthesis 2 cos ⁡ open parenthesis θ close parenthesis sin ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$7 cos^(2) ( θ ) sin^(2) ( θ ) − ( 2 cos ( θ ) sin ( θ ) )^(2)$] For Problems 29–34, simplify the expression, then evaluate. #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis plus 2 cos ⁡ open parenthesis t close parenthesis sin ⁡ open parenthesis t close parenthesis minus 3 cos ⁡ open parenthesis t close parenthesis ,")[$cos ( t ) + 2 cos ( t ) sin ( t ) − 3 cos ( t ) ,$] for #math.equation(block: false, alt: "t equals 143 degrees")[$t = 143^(∘)$] #math.equation(block: true, alt: "minus 2 cos ⁡ open parenthesis t close parenthesis plus 2 cos ⁡ open parenthesis t close parenthesis sin ⁡ open parenthesis t close parenthesis ; 0.6360")[$− 2 cos ( t ) + 2 cos ( t ) sin ( t ) ; " " 0.6360$] #math.equation(block: false, alt: "11 tan ⁡ open parenthesis w close parenthesis minus 4 tan ⁡ open parenthesis w close parenthesis plus 6 tan ⁡ open parenthesis w close parenthesis cos ⁡ open parenthesis w close parenthesis ,")[$11 tan ( w ) − 4 tan ( w ) + 6 tan ( w ) cos ( w ) ,$] for #math.equation(block: false, alt: "w equals 8 degrees")[$w = 8^(∘)$] #math.equation(block: false, alt: "7 tan ⁡ open parenthesis θ close parenthesis minus 4 tan ⁡ open parenthesis ϕ close parenthesis plus 3 tan ⁡ open parenthesis ϕ close parenthesis minus 6 tan ⁡ open parenthesis θ close parenthesis ,")[$7 tan ( θ ) − 4 tan ( ϕ ) + 3 tan ( ϕ ) − 6 tan ( θ ) ,$] for #math.equation(block: false, alt: "θ equals 21 degrees , ϕ equals 89 degrees")[$θ = 21^(∘) , " " ϕ = 89^(∘)$] #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis minus tan ⁡ open parenthesis ϕ close parenthesis ; minus 56.91")[$tan ( θ ) − tan ( ϕ ) ; " " − 56.91$] #math.equation(block: false, alt: "5 sin ⁡ open parenthesis A close parenthesis plus sin ⁡ open parenthesis B close parenthesis minus 6 sin ⁡ open parenthesis B close parenthesis plus 6 sin ⁡ open parenthesis A close parenthesis ,")[$5 sin ( A ) + sin ( B ) − 6 sin ( B ) + 6 sin ( A ) ,$] for #math.equation(block: false, alt: "A equals 111 degrees , B equals 26 degrees")[$A = 111^(∘) , " " B = 26^(∘)$] #math.equation(block: false, alt: "minus sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis x close parenthesis minus sin ⁡ open parenthesis 2 x close parenthesis plus 3 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis x close parenthesis ,")[$− sin ( x ) cos ( x ) − sin ( 2 x ) + 3 sin ( x ) cos ( x ) ,$] for #math.equation(block: false, alt: "x equals 107 degrees")[$x = 107^(∘)$] #math.equation(block: true, alt: "2 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis x close parenthesis minus 2 sin ⁡ open parenthesis 2 x close parenthesis ; 0")[$2 sin ( x ) cos ( x ) − 2 sin ( 2 x ) ; " " 0$] #math.equation(block: false, alt: "4 cos ⁡ open parenthesis u close parenthesis sin ⁡ open parenthesis u close parenthesis plus 3 sin ⁡ open parenthesis 2 u close parenthesis minus 10 sin ⁡ open parenthesis u close parenthesis cos ⁡ open parenthesis u close parenthesis ,")[$4 cos ( u ) sin ( u ) + 3 sin ( 2 u ) − 10 sin ( u ) cos ( u ) ,$] for #math.equation(block: false, alt: "u equals 2 degrees")[$u = 2^(∘)$] For Problems 35–44, decide whether or not the expressions are equivalent. Explain. #math.equation(block: true, alt: "cos ⁡ open parenthesis x plus y close parenthesis")[$cos ( x + y )$], #math.equation(block: true, alt: "cos ⁡ open parenthesis x close parenthesis plus cos ⁡ open parenthesis y close parenthesis")[$" " cos ( x ) + cos ( y )$] No #math.equation(block: true, alt: "sin ⁡ open parenthesis θ minus ϕ close parenthesis")[$sin ( θ − ϕ )$], #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis minus sin ⁡ open parenthesis ϕ close parenthesis")[$" " sin ( θ ) − sin ( ϕ )$] #math.equation(block: true, alt: "tan ⁡ open parenthesis 2 A close parenthesis")[$tan ( 2 A )$], #math.equation(block: true, alt: "2 tan ⁡ open parenthesis A close parenthesis")[$" " 2 tan ( A )$] No #math.equation(block: true, alt: "cos ⁡ open parenthesis the fraction 1 over 2 β close parenthesis")[$cos ( frac(1, 2) β )$], #math.equation(block: true, alt: "the fraction 1 over 2 cos ⁡ open parenthesis β close parenthesis")[$" " frac(1, 2) cos ( β )$] #math.equation(block: true, alt: "open parenthesis sin ⁡ α close parenthesis squared")[$( sin α )^(2)$], #math.equation(block: true, alt: "sin squared ⁡ open parenthesis α close parenthesis")[$sin^(2) ( α )$] Yes #math.equation(block: true, alt: "open parenthesis tan ⁡ open parenthesis B close parenthesis close parenthesis squared")[$( tan ( B ) )^(2)$], #math.equation(block: true, alt: "tan ⁡ open parenthesis B squared close parenthesis")[$tan ( B^(2) )$] #math.equation(block: true, alt: "sin ⁡ open parenthesis 3 t close parenthesis plus sin ⁡ open parenthesis 5 t close parenthesis")[$sin ( 3 t ) + sin ( 5 t )$], #math.equation(block: true, alt: "sin ⁡ open parenthesis 8 t close parenthesis")[$sin ( 8 t )$] No #math.equation(block: true, alt: "3 cos ⁡ open parenthesis 2 x close parenthesis plus 5 cos ⁡ open parenthesis 2 x close parenthesis")[$3 cos ( 2 x ) + 5 cos ( 2 x )$], #math.equation(block: true, alt: "8 cos ⁡ open parenthesis 2 x close parenthesis")[$8 cos ( 2 x )$] #math.equation(block: true, alt: "tan ⁡ open parenthesis θ plus 45 degrees close parenthesis minus tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ + 45^(∘) ) − tan ( θ )$], #math.equation(block: true, alt: "tan ⁡ open parenthesis 45 degrees close parenthesis")[$tan ( 45^(∘) )$] No #math.equation(block: true, alt: "sin ⁡ open parenthesis 30 degrees plus z close parenthesis plus sin ⁡ open parenthesis 30 degrees minus z close parenthesis")[$sin ( 30^(∘) + z ) + sin ( 30^(∘) − z )$], #math.equation(block: true, alt: "sin ⁡ open parenthesis 60 degrees close parenthesis")[$sin ( 60^(∘) )$] For Problems 45–56, multiply or expand. + #math.equation(block: false, alt: "x open parenthesis 2 x minus 1 close parenthesis")[$x ( 2 x − 1 )$] + #math.equation(block: false, alt: "sin ⁡ open parenthesis A close parenthesis open parenthesis 2 sin ⁡ open parenthesis A close parenthesis minus 1 close parenthesis")[$sin ( A ) ( 2 sin ( A ) − 1 )$] + #math.equation(block: false, alt: "2 x squared minus x")[$2 x^(2) − x$] + #math.equation(block: false, alt: "2 sin squared ⁡ open parenthesis A close parenthesis minus sin ⁡ open parenthesis A close parenthesis")[$2 sin^(2) ( A ) − sin ( A )$] + #math.equation(block: false, alt: "q open parenthesis 5 r minus q close parenthesis")[$q ( 5 r − q )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis open parenthesis 5 sin ⁡ open parenthesis θ close parenthesis minus cos ⁡ open parenthesis θ close parenthesis close parenthesis")[$cos ( θ ) ( 5 sin ( θ ) − cos ( θ ) )$] + #math.equation(block: false, alt: "a open parenthesis b minus 3 a close parenthesis")[$a ( b − 3 a )$] + #math.equation(block: false, alt: "tan ⁡ open parenthesis A close parenthesis open parenthesis tan ⁡ open parenthesis B close parenthesis minus 3 tan ⁡ open parenthesis A close parenthesis close parenthesis")[$tan ( A ) ( tan ( B ) − 3 tan ( A ) )$] + #math.equation(block: false, alt: "a b minus 3 a squared")[$a b − 3 a^(2)$] + #math.equation(block: false, alt: "tan ⁡ open parenthesis A close parenthesis tan ⁡ open parenthesis B close parenthesis minus 3 tan squared ⁡ open parenthesis A close parenthesis")[$tan ( A ) tan ( B ) − 3 tan^(2) ( A )$] + #math.equation(block: false, alt: "3 w open parenthesis 2 w minus z close parenthesis")[$3 w ( 2 w − z )$] + #math.equation(block: false, alt: "3 sin ⁡ open parenthesis α close parenthesis open parenthesis 2 sin ⁡ open parenthesis α close parenthesis minus sin ⁡ open parenthesis β close parenthesis close parenthesis")[$3 sin ( α ) ( 2 sin ( α ) − sin ( β ) )$] + #math.equation(block: false, alt: "open parenthesis C plus 1 close parenthesis open parenthesis 2 C minus 1 close parenthesis")[$( C + 1 ) ( 2 C − 1 )$] + #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis ϕ close parenthesis plus 1 close parenthesis open parenthesis 2 cos ⁡ open parenthesis ϕ close parenthesis minus 1 close parenthesis")[$( cos ( ϕ ) + 1 ) ( 2 cos ( ϕ ) − 1 )$] + #math.equation(block: false, alt: "2 C squared plus C minus 1")[$2 C^(2) + C − 1$] + #math.equation(block: false, alt: "2 cos squared ⁡ open parenthesis ϕ close parenthesis plus cos ⁡ open parenthesis ϕ close parenthesis minus 1")[$2 cos^(2) ( ϕ ) + cos ( ϕ ) − 1$] + #math.equation(block: false, alt: "open parenthesis 3 S minus 2 close parenthesis open parenthesis S plus 1 close parenthesis")[$( 3 S − 2 ) ( S + 1 )$] + #math.equation(block: false, alt: "open parenthesis 3 sin ⁡ open parenthesis B close parenthesis minus 2 close parenthesis open parenthesis sin ⁡ open parenthesis B close parenthesis plus 1 close parenthesis")[$( 3 sin ( B ) − 2 ) ( sin ( B ) + 1 )$] + #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis")[$( a + b ) ( a − b )$] + #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis θ close parenthesis plus cos ⁡ open parenthesis ϕ close parenthesis close parenthesis open parenthesis cos ⁡ open parenthesis θ close parenthesis minus cos ⁡ open parenthesis ϕ close parenthesis close parenthesis")[$( cos ( θ ) + cos ( ϕ ) ) ( cos ( θ ) − cos ( ϕ ) )$] + #math.equation(block: false, alt: "a squared minus b squared")[$a^(2) − b^(2)$] + #math.equation(block: false, alt: "cos squared ⁡ open parenthesis θ close parenthesis minus cos squared ⁡ open parenthesis ϕ close parenthesis")[$cos^(2) ( θ ) − cos^(2) ( ϕ )$] + #math.equation(block: false, alt: "open parenthesis t minus w close parenthesis open parenthesis t plus 4 w close parenthesis")[$( t − w ) ( t + 4 w )$] + #math.equation(block: false, alt: "open parenthesis tan ⁡ open parenthesis α close parenthesis minus tan ⁡ open parenthesis β close parenthesis close parenthesis open parenthesis tan ⁡ open parenthesis α close parenthesis plus 4 tan ⁡ open parenthesis β close parenthesis close parenthesis")[$( tan ( α ) − tan ( β ) ) ( tan ( α ) + 4 tan ( β ) )$] + #math.equation(block: false, alt: "open parenthesis 1 minus T close parenthesis squared")[$( 1 − T )^(2)$] + #math.equation(block: false, alt: "open parenthesis 1 minus tan ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$( 1 − tan ( θ ) )^(2)$] + #math.equation(block: false, alt: "1 minus 2 T plus T squared")[$1 − 2 T + T^(2)$] + #math.equation(block: false, alt: "1 minus 2 tan ⁡ open parenthesis θ close parenthesis plus tan squared ⁡ open parenthesis θ close parenthesis")[$1 − 2 tan ( θ ) + tan^(2) ( θ )$] + #math.equation(block: false, alt: "open parenthesis 2 plus 3 S close parenthesis squared")[$( 2 + 3 S )^(2)$] + #math.equation(block: false, alt: "open parenthesis 2 plus 3 sin ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$( 2 + 3 sin ( θ ) )^(2)$] + #math.equation(block: false, alt: "open parenthesis T squared plus 2 close parenthesis open parenthesis T squared minus 2 close parenthesis")[$( T^(2) + 2 ) ( T^(2) − 2 )$] + #math.equation(block: false, alt: "open parenthesis tan squared ⁡ open parenthesis θ close parenthesis plus 2 close parenthesis open parenthesis tan squared ⁡ open parenthesis θ close parenthesis minus 2 close parenthesis")[$( tan^(2) ( θ ) + 2 ) ( tan^(2) ( θ ) − 2 )$] + #math.equation(block: false, alt: "T to the power 4 minus 4")[$T^(4) − 4$] + #math.equation(block: false, alt: "tan to the power 4 ⁡ open parenthesis θ close parenthesis minus 4")[$tan^(4) ( θ ) − 4$] + #math.equation(block: false, alt: "open parenthesis 2 c squared minus 3 close parenthesis open parenthesis 2 c squared plus 3 close parenthesis")[$( 2 c^(2) − 3 ) ( 2 c^(2) + 3 )$] + #math.equation(block: false, alt: "open parenthesis 2 cos squared ⁡ open parenthesis ϕ close parenthesis minus 3 close parenthesis open parenthesis 2 cos squared ⁡ open parenthesis ϕ close parenthesis plus 3 close parenthesis")[$( 2 cos^(2) ( ϕ ) − 3 ) ( 2 cos^(2) ( ϕ ) + 3 )$] For Problems 57–70, factor. + #math.equation(block: false, alt: "9 m plus 15 n")[$9 m + 15 n$] + #math.equation(block: false, alt: "9 cos ⁡ open parenthesis α close parenthesis plus 15 cos ⁡ open parenthesis β close parenthesis")[$9 cos ( α ) + 15 cos ( β )$] + #math.equation(block: false, alt: "3 open parenthesis 3 m plus 5 n close parenthesis")[$3 ( 3 m + 5 n )$] + #math.equation(block: false, alt: "3 open parenthesis 3 cos ⁡ open parenthesis α close parenthesis plus 5 cos ⁡ open parenthesis β close parenthesis close parenthesis")[$3 ( 3 cos ( α ) + 5 cos ( β ) )$] + #math.equation(block: false, alt: "12 p minus 20 q")[$12 p − 20 q$] + #math.equation(block: false, alt: "12 sin ⁡ open parenthesis θ close parenthesis minus 20 sin ⁡ open parenthesis ϕ close parenthesis")[$12 sin ( θ ) − 20 sin ( ϕ )$] + #math.equation(block: false, alt: "5 r squared minus 10 q r")[$5 r^(2) − 10 q r$] + #math.equation(block: false, alt: "5 tan squared ⁡ open parenthesis C close parenthesis minus 10 tan ⁡ open parenthesis B close parenthesis tan ⁡ open parenthesis C close parenthesis")[$5 tan^(2) ( C ) − 10 tan ( B ) tan ( C )$] + #math.equation(block: false, alt: "5 r open parenthesis r minus 2 q close parenthesis")[$5 r ( r − 2 q )$] + #math.equation(block: false, alt: "5 tan ⁡ open parenthesis C close parenthesis open parenthesis tan ⁡ open parenthesis C close parenthesis minus 2 tan ⁡ open parenthesis B close parenthesis close parenthesis")[$5 tan ( C ) ( tan ( C ) − 2 tan ( B ) )$] + #math.equation(block: false, alt: "2 x squared minus 6 x y")[$2 x^(2) − 6 x y$] + #math.equation(block: false, alt: "2 sin squared ⁡ open parenthesis A close parenthesis minus 6 sin ⁡ open parenthesis A close parenthesis cos ⁡ open parenthesis A close parenthesis")[$2 sin^(2) ( A ) − 6 sin ( A ) cos ( A )$] + #math.equation(block: false, alt: "9 C squared minus 1")[$9 C^(2) − 1$] + #math.equation(block: false, alt: "9 cos squared ⁡ open parenthesis β close parenthesis minus 1")[$9 cos^(2) ( β ) − 1$] + #math.equation(block: false, alt: "open parenthesis 3 C plus 1 close parenthesis open parenthesis 3 C minus 1 close parenthesis")[$( 3 C + 1 ) ( 3 C − 1 )$] + #math.equation(block: false, alt: "open parenthesis 3 cos ⁡ open parenthesis β close parenthesis plus 1 close parenthesis open parenthesis 3 cos ⁡ open parenthesis β close parenthesis minus 1 close parenthesis")[$( 3 cos ( β ) + 1 ) ( 3 cos ( β ) − 1 )$] + #math.equation(block: false, alt: "25 S squared minus 16 S")[$25 S^(2) − 16 S$] + #math.equation(block: false, alt: "25 sin squared ⁡ open parenthesis β close parenthesis minus 16 sin ⁡ open parenthesis β close parenthesis")[$25 sin^(2) ( β ) − 16 sin ( β )$] + #math.equation(block: false, alt: "6 T cubed minus 8 T squared")[$6 T^(3) − 8 T^(2)$] + #math.equation(block: false, alt: "6 tan cubed ⁡ open parenthesis A close parenthesis minus 8 tan squared ⁡ open parenthesis A close parenthesis")[$6 tan^(3) ( A ) − 8 tan^(2) ( A )$] + #math.equation(block: false, alt: "2 T squared open parenthesis 3 T minus 4 close parenthesis")[$2 T^(2) ( 3 T − 4 )$] + #math.equation(block: false, alt: "2 tan squared ⁡ open parenthesis A close parenthesis open parenthesis 3 tan ⁡ open parenthesis A close parenthesis minus 4 close parenthesis")[$2 tan^(2) ( A ) ( 3 tan ( A ) − 4 )$] + #math.equation(block: false, alt: "9 T squared minus 15 T cubed")[$9 T^(2) − 15 T^(3)$] + #math.equation(block: false, alt: "9 tan squared ⁡ open parenthesis B close parenthesis minus 15 tan cubed ⁡ open parenthesis B close parenthesis")[$9 tan^(2) ( B ) − 15 tan^(3) ( B )$] + #math.equation(block: false, alt: "t squared minus t minus 20")[$t^(2) − t − 20$] + #math.equation(block: false, alt: "tan squared ⁡ open parenthesis θ close parenthesis minus tan ⁡ open parenthesis θ close parenthesis minus 20")[$tan^(2) ( θ ) − tan ( θ ) − 20$] + #math.equation(block: false, alt: "open parenthesis t minus 5 close parenthesis open parenthesis t plus 4 close parenthesis")[$( t − 5 ) ( t + 4 )$] + #math.equation(block: false, alt: "open parenthesis tan ⁡ open parenthesis θ close parenthesis minus 5 close parenthesis open parenthesis tan ⁡ open parenthesis θ close parenthesis plus 4 close parenthesis")[$( tan ( θ ) − 5 ) ( tan ( θ ) + 4 )$] + #math.equation(block: false, alt: "s squared plus 5 s minus 6")[$s^(2) + 5 s − 6$] + #math.equation(block: false, alt: "sin squared ⁡ open parenthesis A close parenthesis plus 5 sin ⁡ open parenthesis A close parenthesis minus 6")[$sin^(2) ( A ) + 5 sin ( A ) − 6$] + #math.equation(block: false, alt: "3 c squared plus 2 c minus 1")[$3 c^(2) + 2 c − 1$] + #math.equation(block: false, alt: "3 cos squared ⁡ open parenthesis B close parenthesis plus 2 cos ⁡ open parenthesis B close parenthesis minus 1")[$3 cos^(2) ( B ) + 2 cos ( B ) − 1$] + #math.equation(block: false, alt: "open parenthesis 3 c minus 1 close parenthesis open parenthesis c plus 1 close parenthesis")[$( 3 c − 1 ) ( c + 1 )$] + #math.equation(block: false, alt: "open parenthesis 3 cos ⁡ open parenthesis B close parenthesis minus 1 close parenthesis open parenthesis cos ⁡ open parenthesis B close parenthesis plus 1 close parenthesis")[$( 3 cos ( B ) − 1 ) ( cos ( B ) + 1 )$] + #math.equation(block: false, alt: "8 S squared minus 6 S plus 1")[$8 S^(2) − 6 S + 1$] + #math.equation(block: false, alt: "8 sin squared ⁡ open parenthesis ϕ close parenthesis minus 6 sin ⁡ open parenthesis ϕ close parenthesis plus 1")[$8 sin^(2) ( ϕ ) − 6 sin ( ϕ ) + 1$] + #math.equation(block: false, alt: "6 S squared minus 5 S minus 1")[$6 S^(2) − 5 S − 1$] + #math.equation(block: false, alt: "6 sin squared ⁡ open parenthesis α close parenthesis minus 5 sin ⁡ open parenthesis α close parenthesis minus 1")[$6 sin^(2) ( α ) − 5 sin ( α ) − 1$] + #math.equation(block: false, alt: "open parenthesis 6 S plus 1 close parenthesis open parenthesis S minus 1 close parenthesis")[$( 6 S + 1 ) ( S − 1 )$] + #math.equation(block: false, alt: "open parenthesis 6 sin ⁡ open parenthesis α close parenthesis plus 1 close parenthesis open parenthesis sin ⁡ open parenthesis α close parenthesis minus 1 close parenthesis")[$( 6 sin ( α ) + 1 ) ( sin ( α ) − 1 )$] + #math.equation(block: false, alt: "T squared minus 4 T minus 12")[$T^(2) − 4 T − 12$] + #math.equation(block: false, alt: "tan squared ⁡ open parenthesis α close parenthesis minus 4 tan ⁡ open parenthesis α close parenthesis minus 12")[$tan^(2) ( α ) − 4 tan ( α ) − 12$]