Answer Key Chapter 7 – Algebra and Trigonometry 2e | OpenStax

7.1 Angles

Figure 1 Graph of a 240-degree angle with a counterclockwise rotation.
$\frac{3\pi}{2}$
$-135^{\circ}$
$\frac{7\pi}{10}$
$\alpha=150^{\circ}$
$\beta=60^{\circ}$
$\frac{7\pi}{6}$
$\frac{215\pi}{18} = 37.525 \text{ units}$
1.88
$\frac{3\pi}{2} \text{ rad/s}$
1655 kilometers per hour

7.2 Right Triangle Trigonometry

$\frac{7}{25}$
$\sin t = \frac{33}{65}, \cos t = \frac{56}{65}, \tan t = \frac{33}{56}, \sec t = \frac{65}{56}, \csc t = \frac{65}{33}, \cot t = \frac{56}{33}$
$\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \tan(\frac{\pi}{4}) = 1, \sec(\frac{\pi}{4}) = \sqrt{2}, \csc(\frac{\pi}{4}) = \sqrt{2}, \cot(\frac{\pi}{4}) = 1$
2
$\text{adjacent}=10; \text{opposite}=10\sqrt{3};$ missing angle is $\frac{\pi}{6}$
About 52 ft

7.3 Unit Circle

$\cos(t)= -\frac{\sqrt{2}}{2}, \sin(t)= \frac{\sqrt{2}}{2}$
$\cos(\pi)= -1, \sin(\pi)= 0$
$\sin(t)= -\frac{7}{25}$
approximately 0.866025403
$\frac{\pi}{3}$
a. $\cos(315^{\circ})= \frac{\sqrt{2}}{2}, \sin(315^{\circ})= -\frac{\sqrt{2}}{2}$ b. $\cos(-\frac{\pi}{6})= \frac{\sqrt{3}}{2}, \sin(-\frac{\pi}{6})= -\frac{1}{2}$
$(\frac{1}{2}, -\frac{\sqrt{3}}{2})$

7.4 The Other Trigonometric Functions

$\sin t= -\frac{\sqrt{2}}{2}, \cos t= \frac{\sqrt{2}}{2}, \tan t= -1, \sec t= \sqrt{2}, \csc t= -\sqrt{2}, \cot t= -1$
$\sin \frac{\pi}{3}= \frac{\sqrt{3}}{2}, \cos \frac{\pi}{3}= \frac{1}{2}, \tan \frac{\pi}{3}= \sqrt{3}, \sec \frac{\pi}{3}= 2, \csc \frac{\pi}{3}= \frac{2\sqrt{3}}{3}, \cot \frac{\pi}{3}= \frac{\sqrt{3}}{3}$
$\sin(-\frac{7\pi}{4})= \frac{\sqrt{2}}{2}, \cos(-\frac{7\pi}{4})= \frac{\sqrt{2}}{2}, \tan(-\frac{7\pi}{4})= 1, \sec(-\frac{7\pi}{4})= \sqrt{2}, \csc(-\frac{7\pi}{4})= \sqrt{2}, \cot(-\frac{7\pi}{4})= 1$
$-\sqrt{3}$
-2
$\sin t$
$\cos t= -\frac{8}{17}, \sin t= \frac{15}{17}, \tan t= -\frac{15}{8}, \csc t= \frac{17}{15}, \cot t= -\frac{8}{15}$
$\sin t= -1, \cos t= 0, \tan t= \text{Undefined}, \sec t= \text{Undefined}, \csc t= -1, \cot t= 0$
$\sec t= \sqrt{2}, \csc t= \sqrt{2}, \tan t= 1, \cot t= 1$
$\approx -2.414$

7.1 Section Exercises

Figure 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.
Figure 4 Graph of a circle with an angle inscribed.
Figure 5 Graph of a circle with a 135 degree angle inscribed.
Figure 6 Graph of a circle with a 2pi/3 radians angle inscribed.
Figure 7 Graph of a circle with 5pi/6 radians angle inscribed.
Figure 8 Graph of a circle with a –pi/10 radians angle inscribed.
$240^{\circ}$

Figure 9 Graph of a circle showing the equivalence of two angles.
$\frac{4\pi}{3}$

Figure 10 Graph of a circle showing the equivalence of two angles.
$\frac{2\pi}{3}$

Figure 11 Graph of a circle showing the equivalence of two angles.
$\frac{7\pi}{2} \approx 11.00 \text{ in}^2$
$\frac{81\pi}{20} \approx 12.72 \text{ cm}^2$
$20^{\circ}$
$60^{\circ}$
$-75^{\circ}$
$\frac{\pi}{2} \text{ radians}$
$-3\pi \text{ radians}$
$\pi \text{ radians}$
$\frac{5\pi}{6} \text{ radians}$
$\frac{5.02\pi}{3} \approx 5.26 \text{ miles}$
$\frac{25\pi}{9} \approx 8.73 \text{ centimeters}$
$\frac{21\pi}{10} \approx 6.60 \text{ meters}$
$104.7198 \text{ cm}^2$
$0.7697 \text{ in}^2$
$250^{\circ}$
$320^{\circ}$
$\frac{4\pi}{3}$
$\frac{8\pi}{9}$
$1320 \text{ rad/min}, 210.085 \text{ RPM}$
$7 \text{ in/s}, 4.77 \text{ RPM}, 28.65 \text{ deg/s}$
$1,809,557.37 \text{ mm/min} = 30.16 \text{ m/s}$
$5.76 \text{ miles}$
$120^{\circ}$
794 miles per hour
2,234 miles per hour
11.5 inches

7.2 Section Exercises

Figure 12 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.
$\frac{\pi}{6}$
$\frac{\pi}{4}$
$b=\frac{20\sqrt{3}}{3}, c=\frac{40\sqrt{3}}{3}$
$a=10,000, c=10,000.5$
$b=\frac{5\sqrt{3}}{3}, c=\frac{10\sqrt{3}}{3}$
$\frac{5\sqrt{29}}{29}$
$\frac{5\sqrt{2}}{2}$
$\frac{\sqrt{29}}{2}$
$\frac{5\sqrt{41}}{41}$
$\frac{5}{4}$
$\frac{\sqrt{41}}{4}$
$c=14, b=7\sqrt{3}$
$a=15, b=15$
$b=9.9970, c=12.2041$
$a=2.0838, b=11.8177$
$a=55.9808, c=57.9555$
$a=46.6790, b=17.9184$
$a=16.4662, c=16.8341$
188.3159
200.6737
498.3471 ft
1060.09 ft
27.372 ft
22.6506 ft
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.
I
IV
$\frac{\sqrt{3}}{2}$
$\frac{1}{2}$
$\sqrt{2}$
0
-1
$\frac{\sqrt{3}}{2}$
$60^{\circ}$
$80^{\circ}$
$45^{\circ}$
$\frac{\pi}{3}$
$\frac{\pi}{3}$
$\frac{\pi}{8}$
$60^{\circ}$ , Quadrant IV, $\sin(300^{\circ})= -\frac{\sqrt{3}}{2}, \cos(300^{\circ})= \frac{1}{2}$
$45^{\circ}$ , Quadrant II, $\sin(135^{\circ})= \frac{\sqrt{2}}{2}, \cos(135^{\circ})= -\frac{\sqrt{2}}{2}$
$60^{\circ}$ , Quadrant II, $\sin(120^{\circ})= \frac{\sqrt{3}}{2}, \cos(120^{\circ})= -\frac{1}{2}$
$30^{\circ}$ , Quadrant II, $\sin(150^{\circ})= \frac{1}{2}, \cos(150^{\circ})= -\frac{\sqrt{3}}{2}$
$\frac{\pi}{6}$ , Quadrant III, $\sin(\frac{7\pi}{6})= -\frac{1}{2}, \cos(\frac{7\pi}{6})= -\frac{\sqrt{3}}{2}$
$\frac{\pi}{4}$ , Quadrant II, $\sin(\frac{3\pi}{4})= \frac{\sqrt{2}}{2}, \cos(\frac{3\pi}{4})= -\frac{\sqrt{2}}{2}$
$\frac{\pi}{3}$ , Quadrant II, $\sin(\frac{2\pi}{3})= \frac{\sqrt{3}}{2}, \cos(\frac{2\pi}{3})= -\frac{1}{2}$
$\frac{\pi}{4}$ , Quadrant IV, $\sin(\frac{7\pi}{4})= -\frac{\sqrt{2}}{2}, \cos(\frac{7\pi}{4})= \frac{\sqrt{2}}{2}$
$\frac{\sqrt{77}}{9}$
$-\frac{\sqrt{15}}{4}$
$(-10, 10\sqrt{3})$
$(-2.778, 15.757)$
$[-1, 1]$
$\sin t= \frac{1}{2}, \cos t= -\frac{\sqrt{3}}{2}$
$\sin t= -\frac{\sqrt{2}}{2}, \cos t= -\frac{\sqrt{2}}{2}$
$\sin t= \frac{\sqrt{3}}{2}, \cos t= -\frac{1}{2}$
$\sin t= -\frac{\sqrt{2}}{2}, \cos t= \frac{\sqrt{2}}{2}$
$\sin t= 0, \cos t= -1$
$\sin t= -0.596, \cos t= 0.803$
$\sin t= \frac{1}{2}, \cos t= \frac{\sqrt{3}}{2}$
$\sin t= -\frac{1}{2}, \cos t= \frac{\sqrt{3}}{2}$
$\sin t= 0.761, \cos t= -0.649$
$\sin t= 1, \cos t= 0$
-0.1736
0.9511
-0.7071
-0.1392
-0.7660
$\frac{\sqrt{2}}{4}$
$-\frac{\sqrt{6}}{4}$
$\frac{\sqrt{2}}{4}$
$\frac{\sqrt{2}}{4}$
0
$(0, -1)$
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 $\frac{\pi}{4}$ and the terminal side of the angle is in quadrants I and III. Thus, at $x=\frac{\pi}{4}, \frac{5\pi}{4}$ , 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 $\pi$ units.
$\frac{2\sqrt{3}}{3}$
$\sqrt{3}$
$\sqrt{2}$
1
2
$\frac{\sqrt{3}}{3}$
$-\frac{2\sqrt{3}}{3}$
$\sqrt{3}$
$-\sqrt{2}$
–1
-2
$-\frac{\sqrt{3}}{3}$
2
$\frac{\sqrt{3}}{3}$
–2
–1
$\sin t= -\frac{2\sqrt{2}}{3}, \sec t= -3, \csc t= -\frac{3\sqrt{2}}{4}, \tan t= 2\sqrt{2}, \cot t= \frac{\sqrt{2}}{4}$
$\sec t= 2, \csc t= \frac{2\sqrt{3}}{3}, \tan t= \sqrt{3}, \cot t= \frac{\sqrt{3}}{3}$
$-\frac{\sqrt{2}}{2}$
3.1
1.4
$\sin t= \frac{\sqrt{2}}{2}, \cos t= \frac{\sqrt{2}}{2}, \tan t= 1, \cot t= 1, \sec t= \sqrt{2}, \csc t= \sqrt{2}$
$\sin t= -\frac{\sqrt{3}}{2}, \cos t= -\frac{1}{2}, \tan t= \sqrt{3}, \cot t= \frac{\sqrt{3}}{3}, \sec t= -2, \csc t= -\frac{2\sqrt{3}}{3}$
–0.228
–2.414
1.414
1.540
1.556
$\sin t \approx 0.79$
$\csc t \approx 1.16$
even
even
$\frac{\sin t}{\cos t} = \tan t$
13.77 hours, period: $1000\pi$
3.46 inches

Review Exercises

$45^{\circ}$
$-\frac{7\pi}{6}$
10.385 meters
$60^{\circ}$
$\frac{2\pi}{11}$
Figure 13 This is an image of a graph of a circle with a negative angle inscribed.
Figure 14 This is an image of a graph of a circle with an angle inscribed.
1036.73 miles per hour
$\frac{\sqrt{2}}{2}$
$\sqrt{3}$
$72^{\circ}$
$a=\frac{10}{3}, c=\frac{2\sqrt{106}}{3}$
$\frac{6}{\sqrt{11}}$
$a=\frac{5\sqrt{3}}{2}, b=\frac{5}{2}$
369.2136 ft
$\frac{\sqrt{2}}{2}$
$60^{\circ}$
$\frac{\sqrt{3}}{2}$
all real numbers
$\frac{\sqrt{3}}{2}$
$\frac{2\sqrt{3}}{3}$
2
–2.5
$\frac{1}{3}$
cosine, secant

Practice Test

$150^{\circ}$
6.283 centimeters
$15^{\circ}$
Figure 15 This is an image of a graph of a circle with an angle inscribed.
3.351 feet per second, $\frac{2\pi}{75}$ radians per second
$a=\frac{9}{2}, b=\frac{9\sqrt{3}}{2}$
$\frac{1}{2}$
real numbers
1
$-\sqrt{2}$
–0.68
$\frac{\pi}{3}$

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