
# 5.3.3E: Points on Circles Using Sine and Cosine (Exercises)


section 5.3 exercise

1. Find the quadrant in which the terminal point determined by $$t$$ lies if

a. $$\sin (t)<0$$ and $$\cos (t)<0$$
b. $$\sin (t)>0$$ and $$\cos (t)<0$$

2. Find the quadrant in which the terminal point determined by $$t$$ lies if

a. $$\sin (t)<0$$ and $$\cos (t)>0$$
b. $$\sin (t)>0$$ and $$\cos (t)>0$$

3. The point $$P$$ is on the unit circle. If the $$y$$-coordinate of $$P$$ is $$\dfrac{3}{5}$$, and $$P$$ is in quadrant II, find the $$x$$ coordinate.

4. The point $$P$$ is on the unit circle. If the $$x$$-coordinate of $$P$$ is $$\dfrac{1}{5}$$, and $$P$$ is in quadrant IV, find the $$y$$ coordinate.

5. If $${\rm cos}\left(\theta \right)=\dfrac{1}{7}$$ and $$\theta$$ is in the 4$${}^{th}$$ quadrant, find $${\rm sin}\left(\theta \right)$$.

6. If $${\rm cos}\left(\theta \right)=\dfrac{2}{9}$$ and $$\theta$$ is in the 1$${}^{st}$$ quadrant, find $${\rm sin}\left(\theta \right)$$.

7. If $${\rm sin}\left(\theta \right)=\dfrac{3}{8}$$ and $$\theta$$ is in the 2$${}^{nd}$$ quadrant, find $${\rm cos}\left(\theta \right)$$.

8. If $${\rm sin}\left(\theta \right)=-\dfrac{1}{4}$$ and $$\theta$$ is in the 3$${}^{rd}$$ quadrant, find $$\cos \left(\theta \right)$$.

9. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle.

a. 225$$\mathrm{{}^\circ}$$
b. 300$$\mathrm{{}^\circ}$$
c. 135$$\mathrm{{}^\circ}$$
d. 210$$\mathrm{{}^\circ}$$

10. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle.

a. 120$$\mathrm{{}^\circ}$$
b. 315$$\mathrm{{}^\circ}$$
c. 250$$\mathrm{{}^\circ}$$
d. 150$$\mathrm{{}^\circ}$$

11. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle.

a. $$\dfrac{5\pi }{4}$$
b. $$\dfrac{7\pi }{6}$$
c. $$\dfrac{5\pi }{3}$$
d. $$\dfrac{3\pi }{4}$$

12. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle.

a. $$\dfrac{4\pi }{3}$$
b. $$\dfrac{2\pi }{3}$$
c. $$\dfrac{5\pi }{6}$$
d. $$\dfrac{7\pi }{4}$$

13. Give exact values for $${\rm sin}\left(\theta \right)$$ and $${\rm cos}\left(\theta \right)$$ for each of these angles.

a. $$-\dfrac{3\pi }{4}$$
b. $$\dfrac{23\pi }{6}$$
c. $$-\dfrac{\pi }{2}$$
d. $$5\pi$$

14. Give exact values for $${\rm sin}\left(\theta \right)$$ and $${\rm cos}\left(\theta \right)$$ for each of these angles.

a. $$-\dfrac{2\pi }{3}$$
b. $$\dfrac{17\pi }{4}$$
c. $$-\dfrac{\pi }{6}$$
d. $$10\pi$$

15. Find an angle $$\theta$$ with $$0<\theta <360{}^\circ$$ or $$0<\theta <2\pi$$ that has the same sine value as:

a. $$\dfrac{\pi }{3}$$
b. 80$$\mathrm{{}^\circ}$$
c. 140$$\mathrm{{}^\circ}$$
d. $$\dfrac{4\pi }{3}$$
e. 305$$\mathrm{{}^\circ}$$

16. Find an angle $$\theta$$ with $$0<\theta <360{}^\circ$$ or $$0<\theta <2\pi$$ that has the same sine value as:

a. $$\dfrac{\pi }{4}$$
b. 15$$\mathrm{{}^\circ}$$
c. 160$$\mathrm{{}^\circ}$$
d. $$\dfrac{7\pi }{6}$$
e. 340$$\mathrm{{}^\circ}$$

17. Find an angle $$\theta$$ with $$0<\theta <360{}^\circ$$ or $$0<\theta <2\pi$$ that has the same cosine value as:

a. $$\dfrac{\pi }{3}$$
b. 80$$\mathrm{{}^\circ}$$
c. 140$$\mathrm{{}^\circ}$$
d. $$\dfrac{4\pi }{3}$$
e. 305$$\mathrm{{}^\circ}$$

18. 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. $$\dfrac{7\pi }{6}$$
e. 340$$\mathrm{{}^\circ}$$

19. Find the coordinates of the point on a circle with radius 15 corresponding to an angle of 220$$\mathrm{{}^\circ}$$.

20. Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 280$$\mathrm{{}^\circ}$$.

21. Marla is running clockwise around a circular track. She runs at a constant speed of 3 meters per second. She takes 46 seconds to complete one lap of the track. From her starting point, it takes her 12 seconds to reach the northernmost point of the track. Impose a coordinate system with the center of the track at the origin, and the northernmost point on the positive $$y$$-axis. [UW]

a. Give Marla’s coordinates at her starting point.
b. Give Marla’s coordinates when she has been running for 10 seconds.
c. Give Marla’s coordinates when she has been running for 901.3 seconds.

1. a. III
b. II

3. $$-\dfrac{4}{5}$$

5. $$-\dfrac{4\sqrt{3}}{7}$$

7. $$-\dfrac{\sqrt{55}}{8}$$

9. a. reference: $$45^{\circ}$$. Quadrant III. sin($$225^{\circ}$$) = $$-\dfrac{\sqrt{2}}{2}$$. cos($$225^{\circ}$$) = $$-\dfrac{\sqrt{2}}{2}$$
b. reference: $$60^{\circ}$$. Quadrant IV. sin($$300^{\circ}$$) = $$-\dfrac{\sqrt{3}}{2}$$. cos($$300^{\circ}$$) = $$\dfrac{1}{2}$$
c. reference: $$45^{\circ}$$. Quadrant II. sin($$135^{\circ}$$) = $$\dfrac{\sqrt{2}}{2}$$. cos($$135^{\circ}$$) = $$-\dfrac{\sqrt{2}}{2}$$
d. reference: $$30^{\circ}$$. Quadrant III. sin($$210^{\circ}$$) = $$-\dfrac{1}{2}$$. cos($$210^{\circ}$$) = $$-\dfrac{\sqrt{3}}{2}$$

11. a. reference: $$\dfrac{\pi}{4}$$. Quadrant III. sin($$\dfrac{5\pi}{4}$$) = $$-\dfrac{\sqrt{2}}{2}$$. cos($$\dfrac{5\pi}{4}$$) = $$-\dfrac{\sqrt{2}}{2}$$
b. reference: $$\dfrac{\pi}{6}$$. Quadrant III. sin($$\dfrac{7\pi}{6}$$) = $$-\dfrac{1}{2}$$. cos($$\dfrac{7\pi}{6}$$) = $$-\dfrac{\sqrt{3}}{2}$$
c. reference: $$\dfrac{\pi}{3}$$. Quadrant IV. sin($$\dfrac{5\pi}{3}$$) = $$-\dfrac{\sqrt{3}}{2}$$. cos($$\dfrac{5\pi}{3}$$) = $$\dfrac{1}{2}$$
d. reference: $$\dfrac{\pi}{4}$$. Quadrant II. sin(\dfrac{3\pi}{4}\)) = $$\dfrac{\sqrt{2}}{2}$$. cos($$\dfrac{3\pi}{4}$$) = $$-\dfrac{\sqrt{2}}{2}$$

13. a. sin($$-\dfrac{3\pi}{4}$$) = $$-\dfrac{\sqrt{2}}{2}$$ cos($$-\dfrac{3\pi}{4}$$) = $$-\dfrac{\sqrt{2}}{2}$$
b. sin($$\dfrac{23\pi}{6}$$) = $$-\dfrac{1}{2}$$ cos($$\dfrac{23\pi}{6}$$) = $$\dfrac{\sqrt{3}}{2}$$
c. sin($$-\dfrac{\pi}{2}$$) = -1 cos($$-\dfrac{\pi}{2}$$) = 0
d. sin$$(5\pi)$$ = 0 cos$$(5\pi)$$ = -1

15. a. $$\dfrac{2\pi}{3}$$
b. $$100^{\circ}$$
c. $$40^{\circ}$$
d. $$\dfrac{5\pi}{3}$$
e. $$235^{\circ}$$

17. a. $$\dfrac{5\pi}{3}$$
b. $$280^{\circ}$$
c. $$220^{\circ}$$
d. $$\dfrac{2\pi}{3}$$
e. $$55^{\circ}$$

19. (-11.491, -9.642)