#set document(title: "6.4 Chapter 6 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"))


    
== 6.4#h(0.6em)Chapter 6 Summary and Review


    
      
=== Key Concepts


      + The distance we travel around a circle of radius  is proportional to the angle of displacement.
                #math.equation(block: true, alt: "Arclength equals open parenthesis fraction of one revolution close parenthesis times open parenthesis 2 π r close parenthesis")[$"Arclength" " " = " " ( "fraction of one revolution" ) ⋅ ( 2 π r )$]
+ We measure angles in radians when we work with arclength.#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Radians]

The #strong[radian measure] of an angle is given by

#math.equation(block: true, alt: "open parenthesis fraction of one revolution times 2 π close parenthesis")[$( "fraction of one revolution" × 2 π )$]
]
+ An arclength equal to one radius determines a central angle of one radian.
+ Radian measure can be expressed as multiples of #math.equation(block: false, alt: "π")[$π$] or as decimals.#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Degrees], [#math.equation(block: false, alt: "the fraction Radians: over Exact Values")[$display(frac("Radians:", "Exact Values"))$]], [#math.equation(block: false, alt: "the fraction Radians: Decimal over Approximations")[$display(frac("Radians: Decimal", "Approximations"))$]]),
  [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]],
  [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "1.57")[$1.57$]],
  [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "3.14")[$3.14$]],
  [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "4.71")[$4.71$]],
  [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "6.28")[$6.28$]],
))

#figure(figph[Quadrantal angles on unit circle radian approximation], alt: "Quadrantal angles on unit circle radian approximation", caption: none)
+ We multiply by the appropriate conversion factor to convert between degrees and radians.#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Unit Conversion for Angles]

#math.equation(block: true, alt: "the fraction 180 degrees over π radians equals 1")[$display(frac(180^(∘), π " " "radians")) = 1$]
]

 #linebreak() To convert from radians to degrees we multiply the radian measure by #math.equation(block: false, alt: "the fraction 180 degrees over π")[$display(frac(180^(∘), π))$]. #linebreak() To convert from degrees to radians we multiply the degree measure by #math.equation(block: false, alt: "the fraction π over 180")[$display(frac(π, 180))$].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Arclength Formula]

On a circle of radius #math.equation(block: false, alt: "r")[$r$], the #strong[length #math.equation(block: false, alt: "s")[$s$] of an arc] spanned by an angle #math.equation(block: false, alt: "θ")[$θ$] in radians is

#math.equation(block: true, alt: "s equals r θ")[$s = r θ$]
]
+ On a #strong[unit circle], the measure of a (positive) angle in radians is equal to the length of the arc it spans.
+ The sine, cosine, or tangent of a particular angle is the same whether the angle is measured in radians or in degrees.
+ You should be familiar with the trig values of the special angles in radians.#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Degrees], [Radians], [Sine], [Cosine], [Tangent]),
  [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]],
  [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]], [#math.equation(block: false, alt: "the fraction the square root of 3 over 2")[$display(frac(sqrt(3), 2))$]], [#math.equation(block: false, alt: "the fraction 1 over the square root of 3")[$display(frac(1, sqrt(3)))$]],
  [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]], [#math.equation(block: false, alt: "the fraction 1 over the square root of 2")[$display(frac(1, sqrt(2)))$]], [#math.equation(block: false, alt: "the fraction 1 over the square root of 2")[$display(frac(1, sqrt(2)))$]], [#math.equation(block: false, alt: "1")[$1$]],
  [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "the fraction π over 3")[$display(frac(π, 3))$]], [#math.equation(block: false, alt: "the fraction the square root of 3 over 2")[$display(frac(sqrt(3), 2))$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]], [#math.equation(block: false, alt: "the square root of 3")[$sqrt(3)$]],
  [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [undefined],
))
+ To find the sine or cosine of a real number #math.equation(block: false, alt: "t")[$t$], we draw an arc of length #math.equation(block: false, alt: "t")[$t$] on a unit circle, and then find the sine or cosine of the angle #math.equation(block: false, alt: "θ")[$θ$] determined by the arc.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Coordinates on a Unit Circle]

The #strong[coordinates] of the point #math.equation(block: false, alt: "P")[$P$] determined by an arc of length #math.equation(block: false, alt: "t")[$t$] in standard position on a unit circle are

#math.equation(block: true, alt: "open parenthesis x , y close parenthesis equals open parenthesis cos ⁡ open parenthesis t close parenthesis , sin ⁡ open parenthesis t close parenthesis close parenthesis")[$( x , y ) = ( cos ( t ) , sin ( t ) )$]#figure(figph[circle], alt: "circle", caption: none)
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Circular Functions]

Let #math.equation(block: false, alt: "P")[$P$] be the terminal point of an arc of length #math.equation(block: false, alt: "t")[$t$] in standard position on a unit circle. The #strong[circular functions] of #math.equation(block: false, alt: "t")[$t$] are defined by

#math.equation(block: true, alt: "cos ⁡ open parenthesis t close parenthesis, equals x; sin ⁡ open parenthesis t close parenthesis, equals y; tan ⁡ open parenthesis t close parenthesis, equals the fraction y over x , x not equal to 0")[$cos ( t ) & = x \ sin ( t ) & = y \ tan ( t ) & = display(frac(y, x)) , " " " " x ≠ 0$]#figure(figph[circle], alt: "circle", caption: none)
]
+ The #strong[domain] of a function is the set of all possible input values. The #strong[range] of a function is the set of all output values for the function.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals sin ⁡ open parenthesis x close parenthesis")[$f ( x ) = sin ( x )$]]

Domain: all real numbers

Range: #math.equation(block: false, alt: "open bracket minus 1 , 1 close bracket")[$[ − 1 , 1 ]$]

Period: #math.equation(block: false, alt: "2 π")[$2 π$]
]

#figure(figph[sine graph], alt: "sine graph", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals cos ⁡ open parenthesis x close parenthesis")[$g ( x ) = cos ( x )$]]

Domain: all real numbers

Range: #math.equation(block: false, alt: "open bracket minus 1 , 1 close bracket")[$[ − 1 , 1 ]$]

Period: #math.equation(block: false, alt: "2 π")[$2 π$]
]

#figure(figph[cosine graph], alt: "cosine graph", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals tan ⁡ open parenthesis x close parenthesis")[$h ( x ) = tan ( x )$]]

Domain: all real numbers except 
                    #math.equation(block: false, alt: "⋯ , the fraction minus 3 π over 2 , the fraction minus π over 2 , the fraction π over 2 , the fraction 3 π over 2 , ⋯")[$⋯ , " " display(frac(− 3 π, 2)) , " " display(frac(− π, 2)) , " " display(frac(π, 2)) , " " display(frac(3 π, 2)) , " " ⋯$]

Range: all real numbers

Period: #math.equation(block: false, alt: "π")[$π$]
]

#figure(figph[tangent graph], alt: "tangent graph", caption: none)


    
    
      
=== Chapter 6 Review Problems


      For Problems 1–2, convert from degrees to radians. Give exact answers.


      
        + #math.equation(block: false, alt: "75 degrees")[$75^(∘)$]
+ #math.equation(block: false, alt: "210 degrees")[$210^(∘)$]
+ #math.equation(block: false, alt: "340 degrees")[$340^(∘)$]


        
          + #math.equation(block: false, alt: "the fraction 5 π over 12")[$display(frac(5 π, 12))$]
+ #math.equation(block: false, alt: "the fraction 7 π over 6")[$display(frac(7 π, 6))$]
+ #math.equation(block: false, alt: "the fraction 17 π over 9")[$display(frac(17 π, 9))$]


        
      
      
        + #math.equation(block: false, alt: "130 degrees")[$130^(∘)$]
+ #math.equation(block: false, alt: "300 degrees")[$300^(∘)$]
+ #math.equation(block: false, alt: "12 degrees")[$12^(∘)$]


      
      For Problems 3–4, convert from degrees to radians. Round to two decimal places.


      
        + #math.equation(block: false, alt: "27 degrees")[$27^(∘)$]
+ #math.equation(block: false, alt: "142 degrees")[$142^(∘)$]
+ #math.equation(block: false, alt: "218 degrees")[$218^(∘)$]


        
          + #math.equation(block: false, alt: "0.47")[$0.47$]
+ #math.equation(block: false, alt: "2.48")[$2.48$]
+ #math.equation(block: false, alt: "3.80")[$3.80$]


        
      
      
        + #math.equation(block: false, alt: "76 degrees")[$76^(∘)$]
+ #math.equation(block: false, alt: "328 degrees")[$328^(∘)$]
+ #math.equation(block: false, alt: "111 degrees")[$111^(∘)$]


      
      For Problems 5–8, convert from radians to degrees. Round to hundredths if necessary.


      
        + #math.equation(block: false, alt: "the fraction 5 π over 6")[$display(frac(5 π, 6))$]
+ #math.equation(block: false, alt: "the fraction 3 π over 10")[$display(frac(3 π, 10))$]
+ #math.equation(block: false, alt: "the fraction 23 π over 18")[$display(frac(23 π, 18))$]


        
          + #math.equation(block: false, alt: "150 degrees")[$150^(∘)$]
+ #math.equation(block: false, alt: "54 degrees")[$54^(∘)$]
+ #math.equation(block: false, alt: "230 degrees")[$230^(∘)$]


        
      
      
        + #math.equation(block: false, alt: "the fraction 7 π over 4")[$display(frac(7 π, 4))$]
+ #math.equation(block: false, alt: "the fraction 8 π over 15")[$display(frac(8 π, 15))$]
+ #math.equation(block: false, alt: "the fraction 35 π over 20")[$display(frac(35 π, 20))$]


      
      
        + #math.equation(block: false, alt: "2")[$2$]
+ #math.equation(block: false, alt: "3.6")[$3.6$]
+ #math.equation(block: false, alt: "0.8")[$0.8$]


        
          + #math.equation(block: false, alt: "114.59 degrees")[$114.59^(∘)$]
+ #math.equation(block: false, alt: "206.26 degrees")[$206.26^(∘)$]
+ #math.equation(block: false, alt: "45.84 degrees")[$45.84^(∘)$]


        
      
      
        + #math.equation(block: false, alt: "4")[$4$]
+ #math.equation(block: false, alt: "1.2")[$1.2$]
+ #math.equation(block: false, alt: "5.3")[$5.3$]


      
      For Problems 9–10, express each fraction of one revolution as an angle in radians.


      
        + #math.equation(block: false, alt: "the fraction 2 over 3")[$display(frac(2, 3))$]
+ #math.equation(block: false, alt: "the fraction 7 over 12")[$display(frac(7, 12))$]
+ #math.equation(block: false, alt: "the fraction 9 over 8")[$display(frac(9, 8))$]


        
          + #math.equation(block: false, alt: "the fraction 4 π over 3")[$display(frac(4 π, 3))$]
+ #math.equation(block: false, alt: "the fraction 7 π over 6")[$display(frac(7 π, 6))$]
+ #math.equation(block: false, alt: "the fraction 9 π over 4")[$display(frac(9 π, 4))$]


        
      
      
        + #math.equation(block: false, alt: "the fraction 5 over 4")[$display(frac(5, 4))$]
+ #math.equation(block: false, alt: "the fraction 4 over 6")[$display(frac(4, 6))$]
+ #math.equation(block: false, alt: "the fraction 2 over 5")[$display(frac(2, 5))$]


      
      For Problems 11–12, express each angle in radians as a fraction of one revolution.


      
        + #math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]
+ #math.equation(block: false, alt: "the fraction 5 π over 8")[$display(frac(5 π, 8))$]
+ #math.equation(block: false, alt: "the fraction 7 π over 3")[$display(frac(7 π, 3))$]


        
          + #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]
+ #math.equation(block: false, alt: "the fraction 5 over 16")[$display(frac(5, 16))$]
+ #math.equation(block: false, alt: "the fraction 7 over 6")[$display(frac(7, 6))$]


        
      
      
        + #math.equation(block: false, alt: "the fraction 11 π over 6")[$display(frac(11 π, 6))$]
+ #math.equation(block: false, alt: "the fraction 13 π over 4")[$display(frac(13 π, 4))$]
+ #math.equation(block: false, alt: "the fraction π over 9")[$display(frac(π, 9))$]


      
      For Problems 13–14, in which quadrant on a unit circle does an arc with given length lie?


      
        + #math.equation(block: false, alt: "2.15")[$2.15$]
+ #math.equation(block: false, alt: "1.5")[$1.5$]
+ #math.equation(block: false, alt: "6.0")[$6.0$]


        
          + II
+ I
+ IV


        
      
      
        + #math.equation(block: false, alt: "5.4")[$5.4$]
+ #math.equation(block: false, alt: "4.32")[$4.32$]
+ #math.equation(block: false, alt: "3.1")[$3.1$]


      
      
        + The Earth's radius at the equator is 3960 miles. If you travel 150 miles along the equator, what fraction of its circumference have you covered? How many degrees of longitude have you crossed? Convert your answer to radians.
+ Use the arclength formula to calculate the answer to part (a). Do your answers agree?


        
          + #math.equation(block: false, alt: "0.006 , 2.17 , 0.0379")[$0.006 , " " 2.17 , " " 0.0379$]
+ #math.equation(block: false, alt: "0.0379")[$0.0379$]


        
      
      
        A lawn sprinkler has a range of 15 feet, and waters a porion of a circle whose curved edge is 39.27 feet long. Through what angle does the sprinkler turn?


      
      
        Many telecommunications satellites are put into geostationary orbits, so that they have the same period as the rotation of the earth, and hence stay in the same relative position seen from earth. Hundreds of these satellites orbit 22,300 miles above the equator in what is called the Clarke belt, named after Arthur C. Clarke. What is the speed of the satellites? (The radius of the earth is about 4000 miles.)


        
          #math.equation(block: false, alt: "6885")[$6885$] mph


        
      
      
        The planet Neptune is 4504 million kilometers from the Sun. In one Earth year (365 days), it travels a distance of 171.58 million kilometers around its orbit.


        + What fraction of its orbit does Neptune travel in one Earth year? What angle in radians does it traverse in that time?
+ How many days does it take Neptune to complete one orbit around the Sun? What is its speed, in kilometers per day?
+ Earth is 150 million kilometers from the Sun. What is Earth's orbital speed?


      
      For Problems 19–20, evaluate exactly.


      
        + #math.equation(block: false, alt: "cos ⁡ open parenthesis the fraction 2 π over 3 close parenthesis plus sin ⁡ open parenthesis the fraction π over 6 close parenthesis")[$cos ( display(frac(2 π, 3)) ) + sin ( display(frac(π, 6)) )$]
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis the fraction π over 6 close parenthesis minus 3 tan ⁡ open parenthesis the fraction 4 π over 3 close parenthesis")[$tan ( display(frac(π, 6)) ) − 3 tan ( display(frac(4 π, 3)) )$]
+ #math.equation(block: false, alt: "sin squared ⁡ open parenthesis the fraction 5 π over 4 close parenthesis cos ⁡ open parenthesis π close parenthesis")[$sin^(2) ( display(frac(5 π, 4)) ) cos ( π )$]


        
          + #math.equation(block: false, alt: "0")[$0$]
+ #math.equation(block: false, alt: "the fraction minus 8 over the square root of 3")[$display(frac(− 8, sqrt(3)))$]
+ #math.equation(block: false, alt: "the fraction minus 1 over 2")[$display(frac(− 1, 2))$]


        
      
      
        + #math.equation(block: false, alt: "sin ⁡ open parenthesis the fraction 7 π over 6 close parenthesis cos ⁡ open parenthesis the fraction 3 π over 4 close parenthesis")[$sin ( display(frac(7 π, 6)) ) cos ( display(frac(3 π, 4)) )$]
+ #math.equation(block: false, alt: "4 cos ⁡ open parenthesis the fraction 3 π over 2 close parenthesis plus tan ⁡ open parenthesis the fraction 7 π over 4 close parenthesis")[$4 cos ( display(frac(3 π, 2)) ) + tan ( display(frac(7 π, 4)) )$]
+ #math.equation(block: false, alt: "2 tan squared ⁡ open parenthesis the fraction 5 π over 3 close parenthesis sin ⁡ open parenthesis the fraction 4 π over 3 close parenthesis")[$2 tan^(2) ( display(frac(5 π, 3)) ) sin ( display(frac(4 π, 3)) )$]


      
      For Problems 21–22, sketch an arc with the given length in standard position on a unit circle.  Find the coordinates of the terminal point. Round to tenths.


      
        + #math.equation(block: false, alt: "1")[$1$]
+ #math.equation(block: false, alt: "2")[$2$]
+ #math.equation(block: false, alt: "3")[$3$]


        
          + #math.equation(block: false, alt: "open parenthesis 0.5 , 0.8 close parenthesis")[$( 0.5 , 0.8 )$]
+ #math.equation(block: false, alt: "open parenthesis minus 0.4 , 0.9 close parenthesis")[$( − 0.4 , 0.9 )$]
+ #math.equation(block: false, alt: "open parenthesis minus 1.0 , 0.1 close parenthesis")[$( − 1.0 , 0.1 )$]


        
      
      
        + #math.equation(block: false, alt: "4")[$4$]
+ #math.equation(block: false, alt: "5")[$5$]
+ #math.equation(block: false, alt: "6")[$6$]


      
      In Problems 23–24, the circle has radius #math.equation(block: false, alt: "r")[$r$], and its center is the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$].


      
        Find the coordinates of each point in terms of #math.equation(block: false, alt: "α")[$α$].


        + #math.equation(block: false, alt: "P")[$P$]
+ #math.equation(block: false, alt: "Q")[$Q$]
+ #math.equation(block: false, alt: "R")[$R$]
+ #math.equation(block: false, alt: "S")[$S$]


        #figure(figph[circle], alt: "circle", caption: none)


        
          + #math.equation(block: false, alt: "open parenthesis r cos ⁡ open parenthesis α close parenthesis , r sin ⁡ open parenthesis α close parenthesis close parenthesis")[$( r cos ( α ) , r sin ( α ) )$]
+ #math.equation(block: false, alt: "open parenthesis minus r cos ⁡ open parenthesis α close parenthesis , r sin ⁡ open parenthesis α close parenthesis close parenthesis")[$( − r cos ( α ) , r sin ( α ) )$]
+ #math.equation(block: false, alt: "open parenthesis minus r cos ⁡ open parenthesis α close parenthesis , minus r sin ⁡ open parenthesis α close parenthesis close parenthesis")[$( − r cos ( α ) , − r sin ( α ) )$]
+ #math.equation(block: false, alt: "open parenthesis r cos ⁡ open parenthesis α close parenthesis , minus r sin ⁡ open parenthesis α close parenthesis close parenthesis")[$( r cos ( α ) , − r sin ( α ) )$]


        
      
      
        Find the length of each arc in terms of #math.equation(block: false, alt: "α")[$α$].


        + #math.equation(block: false, alt: "O P")[$O P$]
+ #math.equation(block: false, alt: "O Q")[$O Q$]
+ #math.equation(block: false, alt: "P Q")[$P Q$]
+ #math.equation(block: false, alt: "Q R")[$Q R$]


        #figure(figph[circle], alt: "circle", caption: none)


      
      For Problems 25–26, find an exact value for the area of the sector.


      
        With a central angle of #math.equation(block: false, alt: "135 degrees")[$135^(∘)$] in a circle of radius 4 inches.


        
          #math.equation(block: true, alt: "6 π")[$6 π$]


        
      
      
        With a central angle of #math.equation(block: false, alt: "240 degrees")[$240^(∘)$] in a circle of radius 12 centimeters.


      
      For Problems 27–30, fill in the correct inequality symbol.


      
        If #math.equation(block: false, alt: "the fraction π over 2 less than α less than β less than π")[$display(frac(π, 2)) < α < β < π$], then #math.equation(block: false, alt: "cos ⁡ open parenthesis α close parenthesis bar cos ⁡ open parenthesis β close parenthesis")[$cos ( α ) #h(0.167em) underline(#h(1em)) #h(0.167em) cos ( β )$].


        
          #math.equation(block: true, alt: "greater than")[$>$]


        
      
      
        If #math.equation(block: false, alt: "π less than θ less than ϕ less than the fraction 3 π over 2")[$π < θ < ϕ < display(frac(3 π, 2))$], then #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis bar sin ⁡ open parenthesis ϕ close parenthesis")[$sin ( θ ) #h(0.167em) underline(#h(1em)) #h(0.167em) sin ( ϕ )$].


      
      
        If #math.equation(block: false, alt: "the fraction 3 π over 2 less than s less than t less than 2 π")[$display(frac(3 π, 2)) < s < t < 2 π$], then #math.equation(block: false, alt: "tan ⁡ open parenthesis s close parenthesis bar tan ⁡ open parenthesis t close parenthesis")[$tan ( s ) #h(0.167em) underline(#h(1em)) #h(0.167em) tan ( t )$].


        
          #math.equation(block: true, alt: "less than")[$<$]


        
      
      
        If #math.equation(block: false, alt: "the fraction π over 2 less than x less than y less than the fraction 3 π over 2")[$display(frac(π, 2)) < x < y < display(frac(3 π, 2))$], then #math.equation(block: false, alt: "cos ⁡ open parenthesis x close parenthesis bar cos ⁡ open parenthesis y close parenthesis")[$cos ( x ) #h(0.167em) underline(#h(1em)) #h(0.167em) cos ( y )$].


      
      For Problems 31–34, evaluate the function


      
        #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals 12 minus 2.8 sin ⁡ open parenthesis 3.5 t minus 2 close parenthesis")[$f ( t ) = 12 − 2.8 sin ( 3.5 t − 2 )$] for #math.equation(block: false, alt: "t equals 8")[$t = 8$]


        
          #math.equation(block: true, alt: "9.86")[$9.86$]


        
      
      
        #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2.4 plus 6 tan ⁡ open parenthesis the fraction 3 x minus 5 over 4 close parenthesis")[$h ( x ) = 2.4 + 6 tan ( display(frac(3 x − 5, 4)) )$] for #math.equation(block: false, alt: "x equals 1.8")[$x = 1.8$]


      
      
        #math.equation(block: false, alt: "g open parenthesis z close parenthesis equals 0.07 tan ⁡ open parenthesis 0.4 z plus 0.2 close parenthesis minus 1.3")[$g ( z ) = 0.07 tan ( 0.4 z + 0.2 ) − 1.3$] for #math.equation(block: false, alt: "z equals 22")[$z = 22$]


        
          #math.equation(block: true, alt: "minus 1.33")[$− 1.33$]


        
      
      
        #math.equation(block: false, alt: "F open parenthesis s close parenthesis equals minus 1.5 cos ⁡ open parenthesis the fraction s over 8 minus 3 close parenthesis plus 5")[$F ( s ) = − 1.5 cos ( display(frac(s, 8)) − 3 ) + 5$] for #math.equation(block: false, alt: "s equals 6.2")[$s = 6.2$]


      
      For Problems 35–38, find the reference angle in radians.


      
        + #math.equation(block: false, alt: "the fraction 5 π over 6")[$display(frac(5 π, 6))$]
+ #math.equation(block: false, alt: "the fraction 5 π over 4")[$display(frac(5 π, 4))$]
+ #math.equation(block: false, alt: "the fraction 3 π over 8")[$display(frac(3 π, 8))$]
+ #math.equation(block: false, alt: "the fraction 7 π over 12")[$display(frac(7 π, 12))$]


        
          + #math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$]
+ #math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]
+ #math.equation(block: false, alt: "the fraction 3 π over 8")[$display(frac(3 π, 8))$]
+ #math.equation(block: false, alt: "the fraction 5 π over 12")[$display(frac(5 π, 12))$]


        
      
      
        + #math.equation(block: false, alt: "the fraction 5 π over 3")[$display(frac(5 π, 3))$]
+ #math.equation(block: false, alt: "the fraction 9 π over 8")[$display(frac(9 π, 8))$]
+ #math.equation(block: false, alt: "the fraction 17 π over 12")[$display(frac(17 π, 12))$]
+ #math.equation(block: false, alt: "the fraction 7 π over 6")[$display(frac(7 π, 6))$]


      
      
        + #math.equation(block: false, alt: "2.8")[$2.8$]
+ #math.equation(block: false, alt: "3.9")[$3.9$]
+ #math.equation(block: false, alt: "5.03")[$5.03$]
+ #math.equation(block: false, alt: "1.5")[$1.5$]


        
          + #math.equation(block: false, alt: "0.34")[$0.34$]
+ #math.equation(block: false, alt: "0.76")[$0.76$]
+ #math.equation(block: false, alt: "1.25")[$1.25$]
+ #math.equation(block: false, alt: "1.5")[$1.5$]


        
      
      
        + #math.equation(block: false, alt: "1.2")[$1.2$]
+ #math.equation(block: false, alt: "6.2")[$6.2$]
+ #math.equation(block: false, alt: "2.36")[$2.36$]
+ #math.equation(block: false, alt: "4.15")[$4.15$]


      
      For Problems 39–40, find the angle of inclination of the line.


      
        #math.equation(block: true, alt: "2 x plus 5 y equals minus 3")[$2 x + 5 y = − 3$]


        
          #math.equation(block: true, alt: "158.2 degrees")[$158.2^(∘)$]


        
      
      
        #math.equation(block: true, alt: "the fraction x over 8 minus the fraction y over 11 equals 1")[$display(frac(x, 8)) − display(frac(y, 11)) = 1$]


      
      
        + Prepare a Cartesian coordinate system with the #math.equation(block: false, alt: "x")[$x$]-axis scaled from 0 to 10 and the #math.equation(block: false, alt: "y")[$y$]-axis scaled from #math.equation(block: false, alt: "minus 2")[$− 2$] to #math.equation(block: false, alt: "2")[$2$]. Label multiples of #math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$] on the #math.equation(block: false, alt: "x")[$x$]-axis.
+ Sketch an accurate graph of #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$] on the grid. Sketch an accurate graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$y = sin ( x )$] on the same grid.


        
          #figure(figph[graph of cosine and sine], alt: "graph of cosine and sine", caption: none)


        
      
      
        + Use your calculator to make a table of values for #math.equation(block: false, alt: "Y sub 1 equals sin ⁡ open parenthesis x close parenthesis , Y sub 2 equals cos ⁡ open parenthesis x close parenthesis ,")[$Y_(1) = sin ( x ) , " " Y_(2) = cos ( x ) ,$] and #math.equation(block: false, alt: "Y sub s equals the fraction Y sub 1 over Y sub 2")[$Y_(s) = display(frac(Y_(1), Y_(2)))$], for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 3 ,")[$0 ≤ x ≤ 3 ,$] and #math.equation(block: false, alt: "Δ x equals 0.1")[$upright(Δ) x = 0.1$].
+ Plot the points #math.equation(block: false, alt: "open parenthesis x , Y sub 3 close parenthesis")[$( x , Y_(3) )$] from the table. Identify the graph.


      
      For Problems 43–46,


      + Graph the function for #math.equation(block: false, alt: "0 less than or equal to s less than 2 π")[$0 ≤ s < 2 π$]. State the amplitude, period, and midline of the graph.
+ Use the graph to solve the equation for #math.equation(block: false, alt: "0 less than or equal to s less than 2 π")[$0 ≤ s < 2 π$].


      
        + #math.equation(block: false, alt: "h open parenthesis s close parenthesis equals 5 plus 3 cos ⁡ open parenthesis 2 s close parenthesis")[$h ( s ) = 5 + 3 cos ( 2 s )$]
+ #math.equation(block: false, alt: "5 plus 3 cos ⁡ open parenthesis 2 s close parenthesis equals 4.56")[$5 + 3 cos ( 2 s ) = 4.56$]


        
          + #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)

mid: #math.equation(block: false, alt: "y equals 5")[$y = 5$], amp: #math.equation(block: false, alt: "3")[$3$], period: #math.equation(block: false, alt: "π")[$π$]
+ #figure(figph[sinusoidal curve and horizontal line], alt: "sinusoidal curve and horizontal line", caption: none)

#math.equation(block: true, alt: "0.86 , 2.28 , 4.00 , 5.42")[$0.86 , " " 2.28 , " " 4.00 , " " 5.42$]


        
      
      
        + #math.equation(block: false, alt: "f open parenthesis s close parenthesis equals 4 minus 2 sin ⁡ open parenthesis 3 s close parenthesis")[$f ( s ) = 4 − 2 sin ( 3 s )$]
+ #math.equation(block: false, alt: "4 minus 2 sin ⁡ open parenthesis 3 s close parenthesis equals 2")[$4 − 2 sin ( 3 s ) = 2$]


      
      
        + #math.equation(block: false, alt: "g open parenthesis s close parenthesis equals 10 plus 4.8 sin ⁡ open parenthesis s minus 1.5 close parenthesis")[$g ( s ) = 10 + 4.8 sin ( s − 1.5 )$]
+ #math.equation(block: false, alt: "10 plus 4.8 sin ⁡ open parenthesis s minus 1.5 close parenthesis equals 12")[$10 + 4.8 sin ( s − 1.5 ) = 12$]


        
          + #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)

mid: #math.equation(block: false, alt: "y equals 10")[$y = 10$], amp: #math.equation(block: false, alt: "4.8")[$4.8$], period: #math.equation(block: false, alt: "2 π")[$2 π$]
+ #figure(figph[sinusoidal graph and horizontal line], alt: "sinusoidal graph and horizontal line", caption: none)

#math.equation(block: true, alt: "1.93 , 4.2")[$1.93 , " " 4.2$]


        
      
      
        + #math.equation(block: false, alt: "j open parenthesis s close parenthesis equals 1.5 plus 0.25 cos ⁡ open parenthesis s plus 0.5 close parenthesis")[$j ( s ) = 1.5 + 0.25 cos ( s + 0.5 )$]
+ #math.equation(block: false, alt: "1.5 plus 0.25 cos ⁡ open parenthesis s plus 0.5 close parenthesis equals 1.4")[$1.5 + 0.25 cos ( s + 0.5 ) = 1.4$]


      
      For Problems 47–48, solve the equation graphically for #math.equation(block: false, alt: "0 less than or equal to x less than 2 π")[$0 ≤ x < 2 π$].


      
        #math.equation(block: true, alt: "6 plus tan ⁡ open parenthesis x minus the fraction π over 6 close parenthesis equals 7")[$6 + tan ( x − display(frac(π, 6)) ) = 7$]


        
          #math.equation(block: true, alt: "the fraction 5 π over 12 , the fraction 17 π over 12")[$display(frac(5 π, 12)) , " " display(frac(17 π, 12))$]


        
      
      
        #math.equation(block: true, alt: "3 minus tan ⁡ open parenthesis x plus the fraction 3 π over 4 close parenthesis equals 4")[$3 − tan ( x + display(frac(3 π, 4)) ) = 4$]


      
      For Problems 49–52, solve the equation exactly for #math.equation(block: false, alt: "0 less than or equal to x less than 2 π")[$0 ≤ x < 2 π$].


      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals the fraction the square root of 3 over 2")[$sin ( θ ) = display(frac(sqrt(3), 2))$]


        
          #math.equation(block: true, alt: "the fraction π over 3 , the fraction 2 π over 3")[$display(frac(π, 3)) , " " display(frac(2 π, 3))$]


        
      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals minus the fraction 1 over 2")[$sin ( θ ) = − display(frac(1, 2))$]


      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis θ close parenthesis equals minus 1")[$cos ( θ ) = − 1$]


        
          #math.equation(block: true, alt: "π")[$π$]


        
      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis equals the square root of 3")[$tan ( θ ) = sqrt(3)$]


      
      For Problems 53–58, find all solutions between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "2 π")[$2 π$]. Round to two decimal places.


      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis t close parenthesis equals 5")[$tan ( t ) = 5$]


        
          #math.equation(block: true, alt: "1.37 , 4.51")[$1.37 , " " 4.51$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis x close parenthesis equals minus 0.63")[$cos ( x ) = − 0.63$]


      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis h close parenthesis equals minus 0.26")[$sin ( h ) = − 0.26$]


        
          #math.equation(block: true, alt: "6.02 , 3.40")[$6.02 , " " 3.40$]


        
      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis ϕ close parenthesis equals minus 2.5")[$tan ( ϕ ) = − 2.5$]


      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis β close parenthesis equals 0.95")[$cos ( β ) = 0.95$]


        
          #math.equation(block: true, alt: "0.32 , 5.97")[$0.32 , " " 5.97$]


        
      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis α close parenthesis equals 0.1")[$sin ( α ) = 0.1$]


      
      For Problems 59–62, solve for #math.equation(block: false, alt: "x")[$x$].


      
        + #math.equation(block: false, alt: "cos ⁡ open parenthesis x close parenthesis equals 0.35")[$cos ( x ) = 0.35$]
+ #math.equation(block: false, alt: "cos ⁡ open parenthesis 0.35 close parenthesis equals x")[$cos ( 0.35 ) = x$]


        
          + #math.equation(block: false, alt: "1.21 , 5.07")[$1.21 , " " 5.07$]
+ #math.equation(block: false, alt: "0.9394")[$0.9394$]


        
      
      
        + #math.equation(block: false, alt: "sin ⁡ open parenthesis x close parenthesis equals 0.84")[$sin ( x ) = 0.84$]
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis 0.84 close parenthesis equals x")[$sin ( 0.84 ) = x$]


      
      
        + #math.equation(block: false, alt: "sin ⁡ open parenthesis x close parenthesis equals the fraction π over 8")[$sin ( x ) = display(frac(π, 8))$]
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis π 8 close parenthesis equals x")[$sin ( π 8 ) = x$]


        
          + #math.equation(block: false, alt: "0.40 , 2.74")[$0.40 , " " 2.74$]
+ #math.equation(block: false, alt: "0.3827")[$0.3827$]


        
      
      
        + #math.equation(block: false, alt: "tan ⁡ x equals 1.7 π")[$tan x = 1.7 π$]
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis 1.7 π close parenthesis equals x")[$tan ( 1.7 π ) = x$]


      
      For Problems 63–66, sketch the graph. State its domain and range.


      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 2 x squared plus 4")[$g ( x ) = 2 x^(2) + 4$]


        
          #figure(figph[parabola], alt: "parabola", caption: none)


          Dom:  all real numbers, Rge: #math.equation(block: false, alt: "y greater than or equal to 4")[$y ≥ 4$]


        
      
      
        #math.equation(block: true, alt: "h open parenthesis w close parenthesis equals 1 minus the fraction 1 over w squared")[$h ( w ) = 1 − display(frac(1, w^(2)))$]


      
      
        #math.equation(block: true, alt: "F open parenthesis s close parenthesis equals minus the square root of 16 minus s squared")[$F ( s ) = − sqrt(16 − s^(2))$]


        
          #figure(figph[semicircle], alt: "semicircle", caption: none)


          Dom:  #math.equation(block: false, alt: "minus 4 less than or equal to s less than or equal to 4")[$− 4 ≤ s ≤ 4$], Rge: #math.equation(block: false, alt: "minus 4 less than or equal to y less than or equal to 0")[$− 4 ≤ y ≤ 0$]


        
      
      
        #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals 4 plus the square root of 4 minus t")[$G ( t ) = 4 + sqrt(4 − t)$]


      
      
        Prove the Pythagorean identity #math.equation(block: false, alt: "cos squared ⁡ open parenthesis t close parenthesis plus sin squared ⁡ open parenthesis t close parenthesis equals 1")[$cos^(2) ( t ) + sin^(2) ( t ) = 1$] by carrying out the following steps. Sketch a unit circle, and an arc #math.equation(block: false, alt: "t")[$t$] in standard position.


        + Write the equation of the unit circle.
+ Use trig ratios to write the coordinates of the terminal point #math.equation(block: false, alt: "P")[$P$] of the arc.
+ Substitute the coordinates of point #math.equation(block: false, alt: "P")[$P$] into your equation from part (a).
+ Does the identity hold for all values of #math.equation(block: false, alt: "t")[$t$]?


        
          + #math.equation(block: false, alt: "x squared plus y squared equals 1")[$x^(2) + y^(2) = 1$]
+ #math.equation(block: false, alt: "open parenthesis cos ⁡ open parenthesis t close parenthesis , sin ⁡ open parenthesis t close parenthesis close parenthesis")[$( cos ( t ) , sin ( t ) )$]
+ #math.equation(block: false, alt: "cos squared ⁡ open parenthesis t close parenthesis plus sin squared ⁡ open parenthesis t close parenthesis equals 1")[$cos^(2) ( t ) + sin^(2) ( t ) = 1$]
+ Yes


        
      
      
        Prove the tangent identity #math.equation(block: false, alt: "tan ⁡ open parenthesis t close parenthesis equals the fraction sin ⁡ open parenthesis t close parenthesis over cos ⁡ open parenthesis t close parenthesis")[$tan ( t ) = display(frac(sin ( t ), cos ( t )))$] by carrying out the following steps. Sketch an arc #math.equation(block: false, alt: "t")[$t$] in standard position on a unit circle, and label its terminal point #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$].


        + Write #math.equation(block: false, alt: "sin ⁡ open parenthesis t close parenthesis")[$sin ( t )$] and #math.equation(block: false, alt: "cos ⁡ open parenthesis t close parenthesis")[$cos ( t )$] in terms of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$].
+ Write the definition of #math.equation(block: false, alt: "tan ⁡ open parenthesis t close parenthesis")[$tan ( t )$].
+ Substitute your results from part (a) into your expression for (b).
+ Does the identity hold for all values of #math.equation(block: false, alt: "t")[$t$]?