5.1E: An Introduction to the Trigonometric Functions (Exercises)

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Jason Gardner
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Section 5.1 Exercises

Convert each angle to radian measure.
1. 120$\mathrm{{}^\circ}$
2. 412$\mathrm{{}^\circ}$
3. -92$\mathrm{{}^\circ}$
Convert each angle from radians to degrees.
1. $\dfrac{5\pi }{6}$
2. $\dfrac{11\; \pi }{6}$
Find the angle between 0$\mathrm{{}^\circ}$ and 360$\mathrm{{}^\circ}$ that is coterminal to the given angle
1. 685$\mathrm{{}^\circ}$
2. -1400$\mathrm{{}^\circ}$
3. $\dfrac{17\; \pi }{3}$.
4. $-\dfrac{7\; \pi }{6}$
If the point (-3,6) is a point on the terminal side of the angle $\theta$ in standard position, find the exact value of the six trigonometric functions.
If the point (5,-12) is a point on the terminal side of the angle $\theta$ in standard position, find the exact value of the six trigonometric functions.
If $\sin \left(\theta \right)=\dfrac{3}{4}$ and $\theta$ is in quadrant II, find the exact value of the remaining trigonometric functions.
If $\tan \left(\theta \right)=\dfrac{-5}{12}$ and $\theta$ is in quadrant IV, find the exact value of the remaining trigonometric functions.
If $\cos \left(\theta \right)=-\dfrac{1}{3}$, and $\theta$ is in quadrant II, find the exact value of the remaining trigonometric functions.
If $\sin \left(\theta \right)=\dfrac{-1}{5}$ and $\theta$ is in quadrant III, find the exact value of the remaining trigonometric functions.
If $\tan \left(\theta \right)=\dfrac{12}{5}$, and $0\le \theta <\dfrac{\pi }{2}$, find $\sin \left(\theta \right)$.
If $\cos \left(\theta \right)=\dfrac{-3}{7}$, and $\pi \le \theta <\dfrac{3\pi }{2}$, find $\tan \left(\theta \right) $.
If $\sin \left(\theta \right)=\dfrac{4}{9}$, and $\dfrac{\pi }{2} \le \theta < \pi$, find $\cos \left(\theta \right) $.
Find the quadrant in which the terminal point determined by $t$ lies if
1. $\sin (t)<0$ and $\cos (t)<0$
2. $\sin (t)>0$ and $\cos (t)<0$
3. $\tan (t)<0$ and $\sin (t)>0$
Determine whether the statement is possible or not
1. $\sin \left(\theta \right)=2.4$
2. $\cos \left(\theta \right)=-0.98$
For each of the following angles, identify which quadrant the angle lies in, draw the angle, find the reference angle, and draw the reference angle. Then evaluate the sine and cosine of the angle exactly using the reference angle (without technology).
a. 225$\mathrm{{}^\circ}$
b. 300$\mathrm{{}^\circ}$
c. 135$\mathrm{{}^\circ}$
d. 210$\mathrm{{}^\circ}$
For each of the following angles, identify which quadrant the angle lies in, draw the angle, find the reference angle, and draw the reference angle. Then evaluate the sine and cosine of the angle exactly using the reference angle (without technology).
a. $\dfrac{5\pi }{4}$
b. $\dfrac{7\pi }{6}$
c. $\dfrac{5\pi }{3}$
d. $\dfrac{3\pi }{4}$
Give exact values for ${\rm sin}\left(\theta \right)$ and ${\rm cos}\left(\theta \right)$ for each of these angles using the reference angle.
a. $-\dfrac{3\pi }{4}$
b. $\dfrac{23\pi }{6}$
c. $-\dfrac{\pi }{2}$
d. $5\pi$
Find an angle $\theta$ with $0<\theta <360{}^\circ$ or $0<\theta <2\pi$ that has the same value of sine as:
a. $\dfrac{\pi }{3}$
b. 80$\mathrm{{}^\circ}$
c. 140$\mathrm{{}^\circ}$
d. $\dfrac{4\pi }{3}$
Find an angle $\theta$ with $0<\theta <360{}^\circ$ or $0<\theta <2\pi$ that has the same cosine value as:
a. $\dfrac{\pi }{4}$
b. 15$\mathrm{{}^\circ}$
c. 160$\mathrm{{}^\circ}$
d. $\dfrac{7\pi }{6}$
Use a calculator to approximate the values of sine, cosine, tangent, secant of the following values:
a. 0.15 radians
b. 4 radians
c. 70$\mathrm{{}^\circ}$
d. 283$\mathrm{{}^\circ}$

Answer

1. $\dfrac{\pi}{3}$

3. (a) $325^{\circ}$

6. $\cos \left(\theta \right)=\dfrac{\sqrt{7}}{4}$, $\tan \left(\theta \right)=\dfrac{3}{\sqrt{7}}$, $\sec \left(\theta \right)=\dfrac{4}{\sqrt{7}}$, $\csc \left(\theta \right)=\dfrac{4}{3}$, and $\cot \left(\theta \right)=\dfrac{\sqrt{7}}{3}$

10. $\sin \left(\theta \right)=\dfrac{12}{13}$

13. Quadrant III

15. (a) The angle lies in Quadrant III, the reference angle is $325^{\circ}$, $\sin \left(\theta \right)=\dfrac{-\sqrt{2}}{2}$, and $\cos \left(\theta \right)=\dfrac{-\sqrt{2}}{2}$

$a graph of an angle and its reference angle in quadrant III.$

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