13.2.5: Chapter 5

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Try It

5.1 Angles

$Graph of a 240 degree angle.$

$\frac{3 π}{2}$

−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

5.2 Unit Circle: Sine and Cosine Functions

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

$\cos (π) = - 1,$ $\sin (π) = 0$

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

approximately 0.866025403

$\frac{π}{3}$

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

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

5.3 The Other Trigonometric Functions

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

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

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

$- \sqrt{3}$

$- 2$

$\sin t$

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

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

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

10.

$\approx - 2.414$

5.4 Right Triangle Trigonometry

$\frac{7}{25}$

$\begin{array}{l} s i n t = \frac{33}{65}, \cos t = \frac{56}{65}, t a n t = \frac{33}{56}, \\ \sec t = \frac{65}{56}, \csc t = \frac{65}{33}, \cot t = \frac{56}{33} \end{array}$

$\sin (\frac{π}{4}) = \frac{1}{\sqrt{2}}, \cos (\frac{π}{4}) = \frac{1}{\sqrt{2}}, \tan (\frac{π}{4}) = 1,$
$\sec (\frac{π}{4}) = \sqrt{2}, c s c (\frac{π}{4}) = \sqrt{2}, \cot (\frac{π}{4}) = 1$

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

About 52 ft

5.1 Section Exercises

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

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.