14.2.7: Chapter 7

Last updated
Save as PDF

Page ID: 118261

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Try It

7.1 Angles

$Graph of a 240-degree angle with a counterclockwise rotation.$

$\frac{3 π}{2}$

$−135 °$

$\frac{7 π}{10}$

$α = 150°$

$β = 60°$

$\frac{7 π}{6}$

$\frac{215 π}{18} = 37.525 units$

1.88

10.

$\frac{- 3 π}{2}$ rad/s

11.

1655 kilometers per hour

7.2 Right Triangle Trigonometry

$\frac{7}{25}$

$\begin{matrix} sin t & = & \frac{33}{65}, & cos t & = & \frac{56}{65}, & tan t & = & \frac{33}{56}, \\ sec t & = & \frac{65}{56}, & csc t & = & \frac{65}{33}, & cot t & = & \frac{56}{33} \end{matrix}$

$\begin{matrix} sin (\frac{π}{4}) & = & \frac{\sqrt{2}}{2}, cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{π}{4}) = 1, \\ sec (\frac{π}{4}) & = & \sqrt{2}, csc (\frac{π}{4}) = \sqrt{2}, cot (\frac{π}{4}) = 1 \end{matrix}$

$adjacent = 10; opposite = 10 \sqrt{3};$ missing angle is $\frac{π}{6}$

About 52 ft

7.3 Unit Circle

$\cos (t) = - \frac{\sqrt{2}}{2}, \sin (t) = \frac{\sqrt{2}}{2}$

$\cos (π) = - 1, \sin (π) = 0$

$\sin (t) = - \frac{7}{25}$

approximately 0.866025403

$\frac{π}{3}$

ⓐ $cos (315°) = \frac{\sqrt{2}}{2}, sin (315°) = \frac{- \sqrt{2}}{2}$
ⓑ $cos (- \frac{π}{6}) = \frac{\sqrt{3}}{2}, \sin (- \frac{π}{6}) = - \frac{1}{2}$

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$

7.4 The Other Trigonometric Functions

$\sin t = - \frac{\sqrt{2}}{2} \cos t = \frac{\sqrt{2}}{2}, \tan t = - 1, s e c t = \sqrt{2}, \csc t = - \sqrt{2}, \cot t = - 1$

$\sin \frac{π}{3} = \frac{\sqrt{3}}{2}, \cos \frac{π}{3} = \frac{1}{2}, \tan \frac{π}{3} = \sqrt{3}, s e c \frac{π}{3} = 2, c s c \frac{π}{3} = \frac{2 \sqrt{3}}{3}, c o t \frac{π}{3} = \frac{\sqrt{3}}{3}$

$\begin{array}{l} \sin (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \cos (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{- 7 π}{4}) = 1, \\ \sec (\frac{- 7 π}{4}) = \sqrt{2}, \csc (\frac{- 7 π}{4}) = \sqrt{2}, \cot (\frac{- 7 π}{4}) = 1 \end{array}$

$- \sqrt{3}$

$- 2$

$\sin t$

$\begin{array}{l} \cos t = - \frac{8}{17}, \sin t = \frac{15}{17}, \tan t = - \frac{15}{8} \\ \csc t = \frac{17}{15}, \cot t = - \frac{8}{15} \end{array}$

$\begin{array}{l} \sin t = - 1, \cos t = 0, \tan t = Undefined \\ \sec t = Undefined, \csc t = - 1, \cot t = 0 \end{array}$

$\sec t = \sqrt{2}, \csc t = \sqrt{2}, \tan t = 1, \cot t = 1$

10.

$\approx - 2.414$

7.1 Section Exercises

1. $Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.$

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

7. $Graph of a circle with an angle inscribed.$

9. $Graph of a circle with a 135 degree angle inscribed.$

11. $Graph of a circle with a 2pi/3 radians angle inscribed.$

13. $Graph of a circle with 5pi/6 radians angle inscribed.$

15. $Graph of a circle with a –pi/10 radians angle inscribed.$

17.

$240°$

$Graph of a circle showing the equivalence of two angles.$

19.

$\frac{4 π}{3}$

$Graph of a circle showing the equivalence of two angles.$

21.

$\frac{2 π}{3}$

$Graph of a circle showing the equivalence of two angles.$

23.

$\frac{7 π}{2} \approx 11.00 {in}^{2}$

25.

$\frac{81 π}{20} \approx 12.72 {cm}^{2}$

27.

$20°$

29.

$60°$

31.

$−75°$

33.

$\frac{π}{2}$ radians

35.

$−3 π$ radians

37.

$π$ radians

39.

$\frac{5 π}{6}$ radians

41.

$\frac{5.02 π}{3} \approx 5.26$ miles

43.

$\frac{25 π}{9} \approx 8.73$ centimeters

45.

$\frac{21 π}{10} \approx 6.60$ meters

47.

104.7198 cm²

49.

0.7697 in²

51.

$250°$

53.

$320°$

55.

$\frac{4 π}{3}$

57.

$\frac{8 π}{9}$

59.

$1320$ rad/min $210.085$ RPM

61.

$7$ in./s, 4.77 RPM , $28.65$ deg/s

63.

$1, 809, 557.37 mm/min =$
$30.16 m/s$

65.

$5.76$ miles

67.

$120°$

69.

794 miles per hour

71.

2,234 miles per hour

73.

11.5 inches

7.2 Section Exercises

1. $A right triangle with side opposite, adjacent, and hypotenuse labeled.$

The tangent of an angle is the ratio of the opposite side to the adjacent side.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

$\frac{π}{6}$

$\frac{π}{4}$

11.

$b = \frac{20 \sqrt{3}}{3}, c = \frac{40 \sqrt{3}}{3}$

13.

$a = 10,000, c = 10,00.5$

15.

$b = \frac{5 \sqrt{3}}{3}, c = \frac{10 \sqrt{3}}{3}$

17.

$\frac{5 \sqrt{29}}{29}$

19.

$\frac{5}{2}$

21.

$\frac{\sqrt{29}}{2}$

23.

$\frac{5 \sqrt{41}}{41}$

25.

$\frac{5}{4}$

27.

$\frac{\sqrt{41}}{4}$

29.

$c = 14, b = 7 \sqrt{3}$

31.

$a = 15, b = 15$

33.

$b = 9.9970, c = 12.2041$

35.

$a = 2.0838, b = 11.8177$

37.

$a = 55.9808, c = 57.9555$

39.

$a = 46.6790, b = 17.9184$

41.

$a = 16.4662, c = 16.8341$

43.

188.3159

45.

200.6737

47.

498.3471 ft

49.

1060.09 ft

51.

27.372 ft

53.

22.6506 ft

55.

368.7633 ft

7.3 Section Exercises

The unit circle is a circle of radius 1 centered at the origin.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $t,$ formed by the terminal side of the angle $t$ and the horizontal axis.

The sine values are equal.

11.

$\frac{\sqrt{3}}{2}$

13.

$\frac{1}{2}$

15.

$\frac{\sqrt{2}}{2}$

17.

19.

-1

21.

$\frac{\sqrt{3}}{2}$

23.

$60°$

25.

$80 °$

27.

$45°$

29.

$\frac{π}{3}$

31.

$\frac{π}{3}$

33.

$\frac{π}{8}$

35.

$60°,$ Quadrant IV, $sin (300°) = - \frac{\sqrt{3}}{2}$ , $\cos (300°) = \frac{1}{2}$

37.

$45°,$ Quadrant II, $sin (135°) = \frac{\sqrt{2}}{2}$ , $\cos (135°) = - \frac{\sqrt{2}}{2}$

39.

$60°,$ Quadrant II, $sin (120°) = \frac{\sqrt{3}}{2}$ , $\cos (120°) = - \frac{1}{2}$

41.

$30°,$ Quadrant II, $sin (150°) = \frac{1}{2}$ , $\cos (150°) = - \frac{\sqrt{3}}{2}$

43.

$\frac{π}{6},$ Quadrant III, $sin (\frac{7 π}{6}) = - \frac{1}{2}$ , $\cos (\frac{7 π}{6}) = - \frac{\sqrt{3}}{2}$

45.

$\frac{π}{4},$ Quadrant II, $sin (\frac{3 π}{4}) = \frac{\sqrt{2}}{2}$ , $\cos (\frac{3 π}{4}) = - \frac{\sqrt[]{2}}{2}$

47.

$\frac{π}{3},$ Quadrant II, $sin (\frac{2 π}{3}) = \frac{\sqrt{3}}{2}$ , $\cos (\frac{2 π}{3}) = - \frac{1}{2}$

49.

$\frac{π}{4},$ Quadrant IV, $sin (\frac{7 π}{4}) = - \frac{\sqrt{2}}{2}, cos (\frac{7 π}{4}) = \frac{\sqrt{2}}{2}$

51.

$\frac{\sqrt{77}}{9}$

53.

$- \frac{\sqrt{15}}{4}$

55.

$(−10, 10 \sqrt{3})$

57.

$(–2.778, 15.757)$

59.

$[–1, 1]$

61.

$\sin t = \frac{1}{2}, \cos t = - \frac{\sqrt{3}}{2}$

63.

$\sin t = - \frac{\sqrt{2}}{2}, \cos t = - \frac{\sqrt{2}}{2}$

65.

$\sin t = \frac{\sqrt{3}}{2}, \cos t = - \frac{1}{2}$

67.

$\sin t = - \frac{\sqrt{2}}{2}, \cos t = \frac{\sqrt{2}}{2}$

69.

$\sin t = 0, \cos t = - 1$

71.

$\sin t = - 0.596, \cos t = 0.803$

73.

$\sin t = \frac{1}{2}, \cos t = \frac{\sqrt{3}}{2}$

75.

$\sin t = - \frac{1}{2}, \cos t = \frac{\sqrt{3}}{2}$

77.

$\sin t = 0.761, \cos t = - 0.649$

79.

$\sin t = 1, \cos t = 0$

81.

−0.1736

83.

0.9511

85.

−0.7071

87.

−0.1392

89.

−0.7660

91.

$\frac{\sqrt{2}}{4}$

93.

$- \frac{\sqrt{6}}{4}$

95.

$\frac{\sqrt{2}}{4}$

97.

$\frac{\sqrt{2}}{4}$

99.

101.

$(0, –1)$

103.

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

7.4 Section Exercises

Yes, when the reference angle is $\frac{π}{4}$ and the terminal side of the angle is in quadrants I and III. Thus, a $x = \frac{π}{4}, \frac{5 π}{4},$ the sine and cosine values are equal.

Substitute the sine of the angle in for $y$ in the Pythagorean Theorem $x^{2} + y^{2} = 1.$ Solve for $x$ and take the negative solution.

The outputs of tangent and cotangent will repeat every $π$ units.

$\frac{2 \sqrt{3}}{3}$

$\sqrt{3}$

11.

$\sqrt{2}$

13.

15.

17.

$\frac{\sqrt{3}}{3}$

19.

$- \frac{2 \sqrt{3}}{3}$

21.

$\sqrt{3}$

23.

$- \sqrt{2}$

25.

–1

27.

-2

29.

$- \frac{\sqrt{3}}{3}$

31.

33.

$\frac{\sqrt{3}}{3}$

35.

–2

37.

–1

39.

$\sin t = - \frac{2 \sqrt{2}}{3}$ , $\sec t = - 3$ , $\csc t = - \frac{3 \sqrt{2}}{4}$ , $\tan t = 2 \sqrt{2}$ , $\cot t = \frac{\sqrt{2}}{4}$

41.

$\sec t = 2,$ $\csc t = \frac{2 \sqrt{3}}{3},$ $\tan t = \sqrt{3},$ $\cot t = \frac{\sqrt{3}}{3}$

43.

$- \frac{\sqrt{2}}{2}$

45.

3.1

47.

1.4

49.

$\sin t = \frac{\sqrt{2}}{2}$ , $\cos t = \frac{\sqrt{2}}{2}$ , $\tan t = 1$ , $\cot t = 1$ , $\sec t = \sqrt{2}$ , $\csc t = \sqrt{2}$

51.

$\sin t = - \frac{\sqrt{3}}{2}$ , $\cos t = - \frac{1}{2}$ $\tan t = \sqrt{3}$ , $\cot t = \frac{\sqrt{3}}{3}$ , $\sec t = - 2$ , $\csc t = - \frac{2 \sqrt{3}}{3}$

53.

–0.228

55.

–2.414

57.

1.414

59.

1.540

61.

1.556

63.

$\sin (t) \approx 0.79$

65.

$\csc t \approx 1.16$

67.

even

69.

even

71.

$\frac{\sin t}{\cos t} = \tan t$

73.

13.77 hours, period: $1000 π$

75.

3.46 inches

Review Exercises

$45°$

$- \frac{7 π}{6}$

10.385 meters

$60°$

$\frac{2 π}{11}$

11. $This is an image of a graph of a circle with a negative angle inscribed.$

13. $This is an image of a graph of a circle with an angle inscribed.$

15.

1036.73 miles per hour

17.

$\frac{\sqrt{2}}{2}$

19.

$\frac{\sqrt{3}}{3}$

21.

$72°$

23.

$a = \frac{10}{3}, c = \frac{2 \sqrt{106}}{3}$

25.

$\frac{6}{11}$

27.

$a = \frac{5 \sqrt{3}}{2}, b = \frac{5}{2}$

29.

369.2136 ft

31.

$\frac{\sqrt{2}}{2}$

33.

$60°$

35.

$\frac{\sqrt{3}}{2}$

37.

all real numbers

39.

$\frac{\sqrt{3}}{2}$

41.

$\frac{2 \sqrt{3}}{3}$

43.

45.

–2.5

47.

$\frac{1}{3}$

49.

cosine, secant

Practice Test

$150°$

6.283 centimeters

$15°$

7. $This is an image of a graph of a circle with an angle inscribed.$

3.351 feet per second, $\frac{2 π}{75}$ radians per second

11.

$a = \frac{9}{2}, b = \frac{9 \sqrt{3}}{2}$

13.

$\frac{1}{2}$

15.

real numbers

17.

19.

$- \sqrt{2}$

21.

–0.68

23.

$\frac{π}{3}$

Search

Text Color

Text Size

Margin Size

Font Type

Try It

7.1 Angles

7.2 Right Triangle Trigonometry

7.3 Unit Circle

7.4 The Other Trigonometric Functions

7.1 Section Exercises

7.2 Section Exercises

7.3 Section Exercises

7.4 Section Exercises

Review Exercises

Practice Test