#set document(title: "5.4 Chapter 5 Summary and Review", 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")) == 5.4#h(0.6em)Chapter 5 Summary and Review === Key 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. + The parentheses in an expression such as #math.equation(block: false, alt: "sin ⁡ open parenthesis X plus Y close parenthesis")[$sin ( X + Y )$] 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 ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$( cos ( θ ) )^(2)$], 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.) + An #strong[equation] is a statement that two algebraic expressions are equal. It may be true or false. + We can solve equations by trial and error, by using graphs, or by algebraic techniques. + To solve a trigonometric equation, we first isolate the trigonometric ratio on one side of the equation. We then use reference angles to find all the solutions between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$]. + An equation that is true only for certain values of the variable, and false for others, is called a #strong[conditional equation]. An equation that is true for all legitimate values of the variables is called an #strong[identity]. + The expressions on either side of the equal sign in an identity are called #strong[equivalent expressions], because they have the same value for all values of the variable. + We often use identities to replace one form of an expression by a more useful form. + To check to whether an equation is an identity we can compare graphs of #math.equation(block: false, alt: "Y sub 1 equals")[$Y_(1) =$] (left side of the equation) and #math.equation(block: false, alt: "Y sub 2 equals")[$Y_(2) =$] (right side of the equation). If the two graphs agree, the equation is an identity. If the two graphs are not the same, the equation is not an identity. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Pythagorean Identity] For any angle #math.equation(block: false, alt: "θ")[$θ$], #math.equation(block: true, alt: "cos squared ⁡ open parenthesis θ close parenthesis plus sin squared ⁡ open parenthesis θ close parenthesis equals 1")[$cos^(2) ( θ ) + sin^(2) ( θ ) = 1$]Alternate forms: #math.equation(block: true, alt: "cos squared ⁡ open parenthesis θ close parenthesis, equals 1 minus sin squared ⁡ open parenthesis θ close parenthesis; sin squared ⁡ open parenthesis θ close parenthesis, equals 1 minus cos squared ⁡ open parenthesis θ close parenthesis")[$cos^(2) ( θ ) & = 1 − sin^(2) ( θ ) \ sin^(2) ( θ ) & = 1 − cos^(2) ( θ )$] ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Tangent Identity] For any angle not coterminal with #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] or #math.equation(block: false, alt: "270 degrees")[$270^(∘)$], #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis equals the fraction sin ⁡ open parenthesis θ close parenthesis over cos ⁡ open parenthesis θ close parenthesis")[$tan ( θ ) = display(frac(sin ( θ ), cos ( θ )))$] ] + To solve an equation involving more than one trig function, we use identities to rewrite the equation in terms of a single trig function. + To prove an identity, we write one side of the equation in equivalent forms until it is identical to the other side of the equation. === Chapter 5 Review Problems For Problems 1–4, evaluate the expressions for #math.equation(block: false, alt: "x equals 120 degrees , y equals 225 degrees ,")[$x = 120^(∘) , " " y = 225^(∘) ,$] and #math.equation(block: false, alt: "z equals 90 degrees")[$z = 90^(∘)$]. Give exact values for your answers. #math.equation(block: true, alt: "sin squared ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis y close parenthesis")[$sin^(2) ( x ) cos ( y )$] #math.equation(block: true, alt: "the fraction minus 3 over 4 the square root of 2")[$display(frac(− 3, 4 sqrt(2)))$] #math.equation(block: true, alt: "sin ⁡ open parenthesis z close parenthesis minus the fraction 1 over 2 sin ⁡ open parenthesis y close parenthesis")[$sin ( z ) − display(frac(1, 2)) sin ( y )$] #math.equation(block: true, alt: "tan ⁡ open parenthesis z minus x close parenthesis cos ⁡ open parenthesis y minus z close parenthesis")[$tan ( z − x ) cos ( y − z )$] #math.equation(block: true, alt: "the fraction 1 over the square root of 6")[$display(frac(1, sqrt(6)))$] #math.equation(block: true, alt: "the fraction tan squared ⁡ open parenthesis x close parenthesis over 2 cos ⁡ open parenthesis y close parenthesis")[$display(frac(tan^(2) ( x ), 2 cos ( y )))$] For Problems 5–8, evaluate the expressions using a calculator. Are they equal? + #math.equation(block: false, alt: "sin ⁡ open parenthesis 20 degrees plus 40 degrees close parenthesis")[$sin ( 20^(∘) + 40^(∘) )$] + #math.equation(block: false, alt: "sin ⁡ open parenthesis 20 degrees close parenthesis plus sin ⁡ open parenthesis 40 degrees close parenthesis")[$sin ( 20^(∘) ) + sin ( 40^(∘) )$] + #math.equation(block: false, alt: "0.8660")[$0.8660$] + #math.equation(block: false, alt: "0.9848 ;")[$0.9848 ;$] No + #math.equation(block: false, alt: "cos squared ⁡ open parenthesis 70 degrees close parenthesis minus sin squared ⁡ open parenthesis 70 degrees close parenthesis")[$cos^(2) ( 70^(∘) ) − sin^(2) ( 70^(∘) )$] + #math.equation(block: false, alt: "cos ⁡ open parenthesis 2 times 70 degrees close parenthesis")[$cos ( 2 ⋅ 70^(∘) )$] + #math.equation(block: false, alt: "the fraction sin ⁡ open parenthesis 55 degrees close parenthesis over cos ⁡ open parenthesis 55 degrees close parenthesis")[$display(frac(sin ( 55^(∘) ), cos ( 55^(∘) )))$] + #math.equation(block: false, alt: "tan ⁡ open parenthesis 55 degrees close parenthesis")[$tan ( 55^(∘) )$] + #math.equation(block: false, alt: "1.4821")[$1.4821$] + #math.equation(block: false, alt: "1.4821 ;")[$1.4821 ;$] Yes + #math.equation(block: false, alt: "tan ⁡ open parenthesis 80 degrees close parenthesis minus tan ⁡ open parenthesis 10 degrees close parenthesis")[$tan ( 80^(∘) ) − tan ( 10^(∘) )$] + #math.equation(block: false, alt: "tan ⁡ open parenthesis 80 degrees minus 10 degrees close parenthesis")[$tan ( 80^(∘) − 10^(∘) )$] For Problems 9–12, simplify the expression. #math.equation(block: true, alt: "3 sin ⁡ open parenthesis x close parenthesis minus 2 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis y close parenthesis plus 2 sin ⁡ open parenthesis x close parenthesis minus cos ⁡ open parenthesis y close parenthesis")[$3 sin ( x ) − 2 sin ( x ) cos ( y ) + 2 sin ( x ) − cos ( y )$] #math.equation(block: true, alt: "5 sin ⁡ open parenthesis x close parenthesis minus 2 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis y close parenthesis minus cos ⁡ open parenthesis y close parenthesis")[$5 sin ( x ) − 2 sin ( x ) cos ( y ) − cos ( y )$] #math.equation(block: true, alt: "cos ⁡ open parenthesis t close parenthesis plus 3 cos ⁡ open parenthesis 3 t close parenthesis minus 3 cos ⁡ open parenthesis t close parenthesis minus 2 cos ⁡ open parenthesis 3 t close parenthesis")[$cos ( t ) + 3 cos ( 3 t ) − 3 cos ( t ) − 2 cos ( 3 t )$] #math.equation(block: true, alt: "6 tan squared ⁡ open parenthesis θ close parenthesis plus 2 tan ⁡ open parenthesis θ close parenthesis minus open parenthesis 4 tan ⁡ open parenthesis θ close parenthesis close parenthesis squared")[$6 tan^(2) ( θ ) + 2 tan ( θ ) − ( 4 tan ( θ ) )^(2)$] #math.equation(block: true, alt: "2 tan ⁡ open parenthesis θ close parenthesis minus 10 tan squared ⁡ open parenthesis θ close parenthesis")[$2 tan ( θ ) − 10 tan^(2) ( θ )$] #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis open parenthesis 2 cos ⁡ open parenthesis θ close parenthesis minus 2 close parenthesis plus sin ⁡ open parenthesis θ close parenthesis open parenthesis 1 minus sin ⁡ open parenthesis θ close parenthesis close parenthesis")[$sin ( θ ) ( 2 cos ( θ ) − 2 ) + sin ( θ ) ( 1 − sin ( θ ) )$] For Problems 13–16, decide whether or not the expressions are equivalent. Explain. #math.equation(block: true, alt: "cos ⁡ open parenthesis θ close parenthesis plus cos ⁡ open parenthesis 2 θ close parenthesis ; cos ⁡ open parenthesis 3 θ close parenthesis")[$cos ( θ ) + cos ( 2 θ ) ; " " " " cos ( 3 θ )$] Not equivalent #math.equation(block: true, alt: "1 plus sin squared ⁡ open parenthesis x close parenthesis ; open parenthesis 1 plus sin ⁡ open parenthesis x close parenthesis close parenthesis squared")[$1 + sin^(2) ( x ) ; " " " " ( 1 + sin ( x ) )^(2)$] #math.equation(block: true, alt: "3 tan squared ⁡ open parenthesis t close parenthesis minus tan squared ⁡ open parenthesis t close parenthesis ; 2 tan squared ⁡ open parenthesis t close parenthesis")[$3 tan^(2) ( t ) − tan^(2) ( t ) ; " " " " 2 tan^(2) ( t )$] Equivalent #math.equation(block: true, alt: "cos ⁡ open parenthesis 4 θ close parenthesis ; 2 cos ⁡ open parenthesis 2 θ close parenthesis")[$cos ( 4 θ ) ; " " " " 2 cos ( 2 θ )$] For Problems 17–20, multiply or expand. #math.equation(block: true, alt: "open parenthesis cos ⁡ open parenthesis α close parenthesis plus 2 close parenthesis open parenthesis 2 cos ⁡ open parenthesis α close parenthesis minus 3 close parenthesis")[$( cos ( α ) + 2 ) ( 2 cos ( α ) − 3 )$] #math.equation(block: true, alt: "2 cos squared ⁡ α plus cos ⁡ α minus 6")[$2 cos^(2) α + cos α − 6$] #math.equation(block: true, alt: "open parenthesis 1 minus 3 tan ⁡ open parenthesis β close parenthesis close parenthesis squared")[$( 1 − 3 tan ( β ) )^(2)$] #math.equation(block: true, alt: "open parenthesis tan ⁡ open parenthesis ϕ close parenthesis minus cos ⁡ open parenthesis ϕ close parenthesis close parenthesis squared equals 0")[$( tan ( ϕ ) − cos ( ϕ ) )^(2) = 0$] #math.equation(block: true, alt: "tan squared ⁡ open parenthesis ϕ close parenthesis minus 2 tan ⁡ open parenthesis ϕ close parenthesis cos ⁡ open parenthesis ϕ close parenthesis plus cos squared ⁡ open parenthesis ϕ close parenthesis")[$tan^(2) ( ϕ ) − 2 tan ( ϕ ) cos ( ϕ ) + cos^(2) ( ϕ )$] #math.equation(block: true, alt: "open parenthesis sin ⁡ open parenthesis ρ close parenthesis minus 2 cos ⁡ open parenthesis ρ close parenthesis close parenthesis open parenthesis sin ⁡ open parenthesis ρ close parenthesis plus cos ⁡ open parenthesis ρ close parenthesis close parenthesis")[$( sin ( ρ ) − 2 cos ( ρ ) ) ( sin ( ρ ) + cos ( ρ ) )$] For Problems 21–24, factor the expression. #math.equation(block: true, alt: "12 sin ⁡ open parenthesis 3 x close parenthesis minus 6 sin ⁡ open parenthesis 2 x close parenthesis")[$12 sin ( 3 x ) − 6 sin ( 2 x )$] #math.equation(block: true, alt: "6 open parenthesis 2 sin ⁡ open parenthesis 3 x close parenthesis minus sin ⁡ open parenthesis 2 x close parenthesis close parenthesis")[$6 ( 2 sin ( 3 x ) − sin ( 2 x ) )$] #math.equation(block: true, alt: "2 cos squared ⁡ open parenthesis β close parenthesis plus cos ⁡ open parenthesis β close parenthesis")[$2 cos^(2) ( β ) + cos ( β )$] #math.equation(block: true, alt: "1 minus 9 tan squared ⁡ open parenthesis θ close parenthesis")[$1 − 9 tan^(2) ( θ )$] #math.equation(block: true, alt: "open parenthesis 1 plus 3 tan ⁡ open parenthesis θ close parenthesis close parenthesis open parenthesis 1 minus 3 tan ⁡ open parenthesis θ close parenthesis close parenthesis")[$( 1 + 3 tan ( θ ) ) ( 1 − 3 tan ( θ ) )$] #math.equation(block: true, alt: "sin squared ⁡ open parenthesis ϕ close parenthesis minus sin ⁡ open parenthesis ϕ close parenthesis tan ⁡ open parenthesis ϕ close parenthesis minus 2 tan squared ⁡ open parenthesis ϕ close parenthesis")[$sin^(2) ( ϕ ) − sin ( ϕ ) tan ( ϕ ) − 2 tan^(2) ( ϕ )$] For Problems 25–30, reduce the fraction. #math.equation(block: true, alt: "the fraction cos squared ⁡ open parenthesis α close parenthesis minus sin squared ⁡ open parenthesis α close parenthesis over cos ⁡ open parenthesis α close parenthesis minus sin ⁡ open parenthesis α close parenthesis")[$display(frac(cos^(2) ( α ) − sin^(2) ( α ), cos ( α ) − sin ( α )))$] #math.equation(block: true, alt: "cos ⁡ open parenthesis α close parenthesis plus sin ⁡ open parenthesis α close parenthesis")[$cos ( α ) + sin ( α )$] #math.equation(block: true, alt: "the fraction 1 minus tan squared ⁡ open parenthesis θ close parenthesis over 1 minus tan ⁡ open parenthesis θ close parenthesis")[$display(frac(1 − tan^(2) ( θ ), 1 − tan ( θ )))$] #math.equation(block: true, alt: "the fraction 3 cos ⁡ open parenthesis x close parenthesis plus 9 over 2 cos ⁡ open parenthesis x close parenthesis plus 6")[$display(frac(3 cos ( x ) + 9, 2 cos ( x ) + 6))$] #math.equation(block: true, alt: "the fraction 3 over 2")[$display(frac(3, 2))$] #math.equation(block: true, alt: "the fraction 5 sin ⁡ open parenthesis θ close parenthesis minus 10 over sin squared ⁡ open parenthesis θ close parenthesis minus 4")[$display(frac(5 sin ( θ ) − 10, sin^(2) ( θ ) − 4))$] #math.equation(block: true, alt: "the fraction 3 tan squared ⁡ open parenthesis C close parenthesis minus 12 over tan squared ⁡ open parenthesis C close parenthesis minus 4 tan ⁡ open parenthesis C close parenthesis plus 4")[$display(frac(3 tan^(2) ( C ) − 12, tan^(2) ( C ) − 4 tan ( C ) + 4))$] #math.equation(block: true, alt: "the fraction 3 tan ⁡ open parenthesis C close parenthesis plus 2 over tan ⁡ open parenthesis C close parenthesis minus 2")[$display(frac(3 tan ( C ) + 2, tan ( C ) − 2))$] #math.equation(block: true, alt: "the fraction tan squared ⁡ open parenthesis β close parenthesis minus tan ⁡ open parenthesis β close parenthesis minus 6 over tan ⁡ open parenthesis β close parenthesis minus 3")[$display(frac(tan^(2) ( β ) − tan ( β ) − 6, tan ( β ) − 3))$] For Problems 31–32, use a graph to solve the equation for #math.equation(block: false, alt: "0 degrees less than or equal to x less than 360 degrees")[$0^(∘) ≤ x < 360^(∘)$]. Check your solutions by substitution. #math.equation(block: true, alt: "8 cos ⁡ open parenthesis x close parenthesis minus 3 equals 2")[$8 cos ( x ) − 3 = 2$] #math.equation(block: true, alt: "51.32 degrees , 308.68 degrees")[$51.32^(∘) , " " 308.68^(∘)$] #math.equation(block: true, alt: "6 tan ⁡ open parenthesis x close parenthesis minus 2 equals 8")[$6 tan ( x ) − 2 = 8$] For Problems 33–40, find all solutions between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$]. Give exact answers. #math.equation(block: true, alt: "2 cos squared ⁡ open parenthesis θ close parenthesis plus cos ⁡ open parenthesis θ close parenthesis equals 0")[$2 cos^(2) ( θ ) + cos ( θ ) = 0$] #math.equation(block: true, alt: "90 degrees")[$90^(∘)$], #math.equation(block: true, alt: "270 degrees")[$" " 270^(∘)$], #math.equation(block: true, alt: "120 degrees")[$" " 120^(∘)$], #math.equation(block: true, alt: "240 degrees")[$" " 240^(∘)$] #math.equation(block: true, alt: "sin squared ⁡ open parenthesis α close parenthesis minus sin ⁡ open parenthesis α close parenthesis equals 0")[$sin^(2) ( α ) − sin ( α ) = 0$] #math.equation(block: true, alt: "2 sin squared ⁡ open parenthesis x close parenthesis minus sin ⁡ open parenthesis x close parenthesis minus 1 equals 0")[$2 sin^(2) ( x ) − sin ( x ) − 1 = 0$] #math.equation(block: true, alt: "90 degrees")[$90^(∘)$], #math.equation(block: true, alt: "210 degrees")[$" " 210^(∘)$], #math.equation(block: true, alt: "330 degrees")[$" " 330^(∘)$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis B close parenthesis plus 2 cos ⁡ open parenthesis B close parenthesis plus 1 equals 0")[$cos^(2) ( B ) + 2 cos ( B ) + 1 = 0$] #math.equation(block: true, alt: "tan squared ⁡ open parenthesis x close parenthesis equals the fraction 1 over 3")[$tan^(2) ( x ) = display(frac(1, 3))$] #math.equation(block: true, alt: "30 degrees")[$30^(∘)$], #math.equation(block: true, alt: "150 degrees")[$" " 150^(∘)$], #math.equation(block: true, alt: "210 degrees")[$" " 210^(∘)$], #math.equation(block: true, alt: "330 degrees")[$" " 330^(∘)$] #math.equation(block: true, alt: "tan squared ⁡ open parenthesis t close parenthesis minus tan ⁡ open parenthesis t close parenthesis equals 0")[$tan^(2) ( t ) − tan ( t ) = 0$] #math.equation(block: true, alt: "6 cos squared ⁡ open parenthesis α close parenthesis minus 3 cos ⁡ open parenthesis α close parenthesis minus 3 equals 0")[$6 cos^(2) ( α ) − 3 cos ( α ) − 3 = 0$] #math.equation(block: true, alt: "0 degrees")[$0^(∘)$], #math.equation(block: true, alt: "120 degrees")[$" " 120^(∘)$], #math.equation(block: true, alt: "240 degrees")[$" " 240^(∘)$] #math.equation(block: true, alt: "2 sin squared ⁡ open parenthesis θ close parenthesis plus 4 sin ⁡ open parenthesis θ close parenthesis plus 2 equals 0")[$2 sin^(2) ( θ ) + 4 sin ( θ ) + 2 = 0$] For Problems 41–44, solve the equation for #math.equation(block: false, alt: "0 degrees less than or equal to x less than 360 degrees")[$0^(∘) ≤ x < 360^(∘)$]. Round your answers to two decimal places. #math.equation(block: true, alt: "2 minus 5 tan ⁡ open parenthesis θ close parenthesis equals minus 6")[$2 − 5 tan ( θ ) = − 6$] #math.equation(block: true, alt: "57.99 degrees , 237.99 degrees")[$57.99^(∘) , " " 237.99^(∘)$] #math.equation(block: true, alt: "3 plus 5 cos ⁡ open parenthesis θ close parenthesis equals 4")[$3 + 5 cos ( θ ) = 4$] #math.equation(block: true, alt: "3 cos squared ⁡ open parenthesis x close parenthesis plus 7 cos ⁡ open parenthesis x close parenthesis equals 0")[$3 cos^(2) ( x ) + 7 cos ( x ) = 0$] #math.equation(block: true, alt: "90 degrees , 270 degrees")[$90^(∘) , " " 270^(∘)$] #math.equation(block: true, alt: "8 minus 9 sin squared ⁡ open parenthesis x close parenthesis equals 0")[$8 − 9 sin^(2) ( x ) = 0$] A light ray passes from glass to water, with a #math.equation(block: false, alt: "37 degrees")[$37^(∘)$] angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1. #math.equation(block: true, alt: "33.17 degrees")[$33.17^(∘)$] A light ray passes from glass to water, with a #math.equation(block: false, alt: "76 degrees")[$76^(∘)$] angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1. For Problems 47–50, decide which of the following equations are identities. Explain your reasoning. #math.equation(block: true, alt: "cos ⁡ open parenthesis x close parenthesis tan ⁡ open parenthesis x close parenthesis equals sin ⁡ open parenthesis x close parenthesis")[$cos ( x ) tan ( x ) = sin ( x )$] Identity #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals 1 minus cos ⁡ open parenthesis θ close parenthesis")[$sin ( θ ) = 1 − cos ( θ )$] #math.equation(block: true, alt: "tan ⁡ open parenthesis ϕ close parenthesis plus tan ⁡ open parenthesis ϕ close parenthesis equals tan ⁡ open parenthesis 2 ϕ close parenthesis")[$tan ( ϕ ) + tan ( ϕ ) = tan ( 2 ϕ )$] Not an identity #math.equation(block: true, alt: "tan squared ⁡ open parenthesis x close parenthesis equals the fraction sin squared ⁡ open parenthesis x close parenthesis over 1 minus sin squared ⁡ open parenthesis x close parenthesis")[$tan^(2) ( x ) = display(frac(sin^(2) ( x ), 1 − sin^(2) ( x )))$] For Problems 51–54, use graphs to decide which of the following equations are identities. #math.equation(block: true, alt: "cos ⁡ open parenthesis 2 θ close parenthesis equals 2 cos ⁡ open parenthesis θ close parenthesis")[$cos ( 2 θ ) = 2 cos ( θ )$] Not an identity #math.equation(block: true, alt: "cos ⁡ open parenthesis x minus 90 degrees close parenthesis equals sin ⁡ open parenthesis x close parenthesis")[$cos ( x − 90^(∘) ) = sin ( x )$] #math.equation(block: true, alt: "sin ⁡ open parenthesis 2 x close parenthesis equals 2 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis x close parenthesis")[$sin ( 2 x ) = 2 sin ( x ) cos ( x )$] Identity #math.equation(block: true, alt: "cos ⁡ open parenthesis θ plus 90 degrees close parenthesis equals cos ⁡ open parenthesis θ close parenthesis minus 1")[$cos ( θ + 90^(∘) ) = cos ( θ ) − 1$] For Problems 55–58, show that the equation is an identity by transforming the left side into the right side. #math.equation(block: true, alt: "the fraction 1 minus cos squared ⁡ open parenthesis α close parenthesis over tan ⁡ open parenthesis α close parenthesis equals sin ⁡ open parenthesis α close parenthesis cos ⁡ open parenthesis α close parenthesis")[$display(frac(1 − cos^(2) ( α ), tan ( α ))) = sin ( α ) cos ( α )$] #math.equation(block: true, alt: "the fraction 1 minus cos squared ⁡ open parenthesis α close parenthesis over tan ⁡ open parenthesis α close parenthesis equals sin squared ⁡ open parenthesis α close parenthesis times the fraction cos ⁡ open parenthesis α close parenthesis over sin ⁡ open parenthesis α close parenthesis equals sin ⁡ open parenthesis α close parenthesis cos ⁡ open parenthesis α close parenthesis")[$display(frac(1 − cos^(2) ( α ), tan ( α ))) = sin^(2) ( α ) ⋅ display(frac(cos ( α ), sin ( α ))) = sin ( α ) cos ( α )$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis β close parenthesis tan squared ⁡ open parenthesis β close parenthesis plus cos squared ⁡ open parenthesis β close parenthesis equals 1")[$cos^(2) ( β ) tan^(2) ( β ) + cos^(2) ( β ) = 1$] #math.equation(block: true, alt: "the fraction tan ⁡ open parenthesis θ close parenthesis minus sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis over sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$display(frac(tan ( θ ) − sin ( θ ) cos ( θ ), sin ( θ ) cos ( θ ))) = sin ( θ )$] #math.equation(block: true, alt: "the fraction the fraction sin ⁡ open parenthesis θ close parenthesis over cos ⁡ open parenthesis θ close parenthesis minus sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis over sin ⁡ open parenthesis θ close parenthesis times the fraction sin ⁡ open parenthesis θ close parenthesis over cos ⁡ open parenthesis θ close parenthesis, equals the fraction sin ⁡ open parenthesis θ close parenthesis minus sin ⁡ open parenthesis θ close parenthesis cos squared ⁡ open parenthesis θ close parenthesis over sin squared ⁡ open parenthesis θ close parenthesis; equals the fraction sin ⁡ open parenthesis θ close parenthesis open parenthesis 1 minus cos squared ⁡ open parenthesis θ close parenthesis close parenthesis over sin squared ⁡ open parenthesis θ close parenthesis")[$display(frac(display(frac(sin ( θ ), cos ( θ ))) − sin ( θ ) cos ( θ ), sin ( θ ) ⋅ display(frac(sin ( θ ), cos ( θ ))))) & = display(frac(sin ( θ ) − sin ( θ ) cos^(2) ( θ ), sin^(2) ( θ ))) \ & = display(frac(sin ( θ ) ( 1 − cos^(2) ( θ ) ), sin^(2) ( θ ))) = display(frac(sin ( θ ) sin^(2) ( θ ), sin^(2) ( θ ))) = sin ( θ )$] #math.equation(block: true, alt: "tan ⁡ open parenthesis ϕ close parenthesis minus the fraction sin squared ⁡ open parenthesis ϕ close parenthesis over tan ⁡ open parenthesis ϕ close parenthesis equals tan ⁡ open parenthesis ϕ close parenthesis sin squared ⁡ open parenthesis ϕ close parenthesis")[$tan ( ϕ ) − display(frac(sin^(2) ( ϕ ), tan ( ϕ ))) = tan ( ϕ ) sin^(2) ( ϕ )$] For Problems 59–62, simplify, using identities as necessary. #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis plus the fraction cos ⁡ open parenthesis θ close parenthesis over sin ⁡ open parenthesis θ close parenthesis")[$tan ( θ ) + display(frac(cos ( θ ), sin ( θ )))$] #math.equation(block: true, alt: "the fraction 1 over sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis")[$display(frac(1, sin ( θ ) cos ( θ )))$] #math.equation(block: true, alt: "the fraction 1 minus 2 cos squared ⁡ open parenthesis β close parenthesis over sin ⁡ open parenthesis β close parenthesis cos ⁡ open parenthesis β close parenthesis plus the fraction cos ⁡ open parenthesis β close parenthesis over sin ⁡ open parenthesis β close parenthesis")[$display(frac(1 − 2 cos^(2) ( β ), sin ( β ) cos ( β ))) + display(frac(cos ( β ), sin ( β )))$] #math.equation(block: true, alt: "the fraction 1 over 1 minus sin squared ⁡ open parenthesis v close parenthesis minus tan squared ⁡ open parenthesis v close parenthesis")[$display(frac(1, 1 − sin^(2) ( v ))) − tan^(2) ( v )$] #math.equation(block: true, alt: "1")[$1$] #math.equation(block: true, alt: "cos ⁡ open parenthesis u close parenthesis plus open parenthesis sin ⁡ open parenthesis u close parenthesis close parenthesis open parenthesis tan ⁡ open parenthesis u close parenthesis close parenthesis")[$cos ( u ) + ( sin ( u ) ) ( tan ( u ) )$] For Problems 63–66, evaluate the expressions without using a calculator. #math.equation(block: true, alt: "sin ⁡ open parenthesis 137 degrees close parenthesis minus tan ⁡ open parenthesis 137 degrees close parenthesis times cos ⁡ open parenthesis 137 degrees close parenthesis")[$sin ( 137^(∘) ) − tan ( 137^(∘) ) ⋅ cos ( 137^(∘) )$] #math.equation(block: true, alt: "0")[$0$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis 8 degrees close parenthesis plus cos ⁡ open parenthesis 8 degrees close parenthesis times tan ⁡ open parenthesis 8 degrees close parenthesis times sin ⁡ open parenthesis 8 degrees close parenthesis")[$cos^(2) ( 8^(∘) ) + cos ( 8^(∘) ) ⋅ tan ( 8^(∘) ) ⋅ sin ( 8^(∘) )$] #math.equation(block: true, alt: "the fraction 1 over cos squared ⁡ open parenthesis 54 degrees close parenthesis minus tan squared ⁡ open parenthesis 54 degrees close parenthesis")[$display(frac(1, cos^(2) ( 54^(∘) ))) − tan^(2) ( 54^(∘) )$] #math.equation(block: true, alt: "1")[$1$] #math.equation(block: true, alt: "the fraction 2 over cos squared ⁡ open parenthesis 7 degrees close parenthesis minus 2 tan squared ⁡ open parenthesis 7 degrees close parenthesis")[$display(frac(2, cos^(2) ( 7^(∘) ))) − 2 tan^(2) ( 7^(∘) )$] For Problems 67–70,use identities to rewrite each expression. Write #math.equation(block: false, alt: "tan squared ⁡ open parenthesis β close parenthesis plus 1")[$" " tan^(2) ( β ) + 1 " "$] in terms of #math.equation(block: false, alt: "cos squared ⁡ open parenthesis β close parenthesis")[$cos^(2) ( β )$]. #math.equation(block: true, alt: "the fraction 1 over cos squared ⁡ open parenthesis β close parenthesis")[$display(frac(1, cos^(2) ( β )))$] Write #math.equation(block: false, alt: "2 sin squared ⁡ open parenthesis t close parenthesis plus cos ⁡ open parenthesis t close parenthesis")[$" " 2 sin^(2) ( t ) + cos ( t ) " "$] in terms of #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis")[$cos ( t )$]. Write #math.equation(block: false, alt: "the fraction cos ⁡ open parenthesis x close parenthesis over tan ⁡ open parenthesis x close parenthesis")[$" " display(frac(cos ( x ), tan ( x ))) " "$] in terms of #math.equation(block: false, alt: "sin ⁡ open parenthesis x close parenthesis")[$sin ( x )$]. #math.equation(block: true, alt: "sin ⁡ open parenthesis x close parenthesis")[$sin ( x )$] Write #math.equation(block: false, alt: "tan squared ⁡ open parenthesis β close parenthesis plus 1")[$" " tan^(2) ( β ) + 1 " "$] in terms of #math.equation(block: false, alt: "cos squared ⁡ open parenthesis β close parenthesis")[$cos^(2) ( β )$]. For Problems 71–74, find the values of the three trigonometric functions. #math.equation(block: true, alt: "7 tan ⁡ open parenthesis β close parenthesis minus 4 equals 2 , 180 degrees less than β less than 270 degrees")[$7 tan ( β ) − 4 = 2 , " " " " 180^(∘) < β < 270^(∘)$] #math.equation(block: true, alt: "sin ⁡ open parenthesis β close parenthesis equals the fraction minus 6 over the square root of 85 , cos ⁡ open parenthesis β close parenthesis equals the fraction minus 7 over the square root of 85 , tan ⁡ open parenthesis β close parenthesis equals the fraction 6 over 7")[$sin ( β ) = display(frac(− 6, sqrt(85))) , " " cos ( β ) = display(frac(− 7, sqrt(85))) , " " tan ( β ) = display(frac(6, 7))$] #math.equation(block: true, alt: "3 tan ⁡ open parenthesis C close parenthesis plus 5 equals 3 , minus 90 degrees less than C less than 0 degrees")[$3 tan ( C ) + 5 = 3 , " " " " − 90^(∘) < C < 0^(∘)$] #math.equation(block: true, alt: "5 cos ⁡ open parenthesis α close parenthesis plus 3 equals 1 , 90 degrees less than α less than 180 degrees")[$5 cos ( α ) + 3 = 1 , " " " " 90^(∘) < α < 180^(∘)$] #math.equation(block: true, alt: "sin ⁡ open parenthesis α close parenthesis equals the fraction the square root of 21 over 5 , cos ⁡ open parenthesis α close parenthesis equals the fraction minus 2 over 5 , tan ⁡ open parenthesis α close parenthesis equals the fraction minus the square root of 21 over 2")[$sin ( α ) = display(frac(sqrt(21), 5)) , " " cos ( α ) = display(frac(− 2, 5)) , " " tan ( α ) = display(frac(− sqrt(21), 2))$] #math.equation(block: true, alt: "3 sin ⁡ open parenthesis θ close parenthesis plus 2 equals 4 , 90 degrees less than β less than 180 degrees")[$3 sin ( θ ) + 2 = 4 , " " " " 90^(∘) < β < 180^(∘)$] For Problems 75–82, solve the equation for #math.equation(block: false, alt: "0 degrees less than or equal to x less than 360 degrees")[$0^(∘) ≤ x < 360^(∘)$]. Round angles to three decimal places if necessary. #math.equation(block: true, alt: "sin ⁡ open parenthesis w close parenthesis plus 1 equals cos squared ⁡ open parenthesis w close parenthesis")[$sin ( w ) + 1 = cos^(2) ( w )$] #math.equation(block: true, alt: "0 degrees , 180 degrees , 270 degrees")[$0^(∘) , " " 180^(∘) , " " 270^(∘)$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis ϕ close parenthesis minus cos ⁡ open parenthesis ϕ close parenthesis minus sin squared ⁡ open parenthesis ϕ close parenthesis equals 0")[$cos^(2) ( ϕ ) − cos ( ϕ ) − sin^(2) ( ϕ ) = 0$] #math.equation(block: true, alt: "cos ⁡ open parenthesis x close parenthesis plus sin ⁡ open parenthesis x close parenthesis equals 0")[$cos ( x ) + sin ( x ) = 0$] #math.equation(block: true, alt: "135 degrees , 315 degrees")[$135^(∘) , " " 315^(∘)$] #math.equation(block: true, alt: "3 sin ⁡ open parenthesis θ close parenthesis equals the square root of 3 cos ⁡ open parenthesis θ close parenthesis")[$3 sin ( θ ) = sqrt(3) cos ( θ )$] #math.equation(block: true, alt: "2 sin ⁡ open parenthesis β close parenthesis minus tan ⁡ open parenthesis β close parenthesis equals 0")[$2 sin ( β ) − tan ( β ) = 0$] #math.equation(block: true, alt: "0 degrees , 60 degrees , 180 degrees , 300 degrees")[$0^(∘) , " " 60^(∘) , " " 180^(∘) , " " 300^(∘)$] #math.equation(block: true, alt: "6 tan ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis plus 6 equals 0")[$6 tan ( θ ) cos ( θ ) + 6 = 0$] #math.equation(block: true, alt: "cos squared ⁡ open parenthesis t close parenthesis minus sin squared ⁡ open parenthesis t close parenthesis equals 1")[$cos^(2) ( t ) − sin^(2) ( t ) = 1$] #math.equation(block: true, alt: "0 degrees , 180 degrees")[$0^(∘) , " " 180^(∘)$] #math.equation(block: true, alt: "5 cos squared ⁡ open parenthesis β close parenthesis minus 5 sin squared ⁡ open parenthesis β close parenthesis equals minus 5")[$5 cos^(2) ( β ) − 5 sin^(2) ( β ) = − 5$]