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# 5.4.4E: The Other Trigonometric Functions (Exercises)

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Section 5.4 Exercise

1. If $$\theta =\dfrac{\pi \; }{4}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.
2. If $$\theta =\dfrac{7\pi \; }{4}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.
3. If $$\theta =\dfrac{5\pi \; }{6}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.
4. If $$\theta =\dfrac{\pi \; }{6}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
5. If $$\theta =\dfrac{2\pi \; }{3}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.
6. If $$\theta =\dfrac{4\pi \; }{3}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
7. Evaluate: a. $$\sec \left(135{}^\circ \right)$$ b. $$\csc \left(210{}^\circ \right)$$ c. $$\tan \left(60{}^\circ \right)$$ d. $$\cot \left(225{}^\circ \right)$$
8. Evaluate: a. $$\sec \left(30{}^\circ \right)$$ b. $$\csc \left(315{}^\circ \right)$$ c. $$\tan \left(135{}^\circ \right)$$ d. $$\cot \left(150{}^\circ \right)$$
9. If $$\sin \left(\theta \right)=\dfrac{3}{4}$$, and $$\theta$$ is in quadrant II, find $$\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
10. If $$\sin \left(\theta \right)=\dfrac{2}{7}$$, and $$\theta$$ is in quadrant II, find $$\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
11. If $$\cos \left(\theta \right)=-\dfrac{1}{3}$$, and $$\theta$$ is in quadrant III, find $$\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
12. If $$\cos \left(\theta \right)=\dfrac{1}{5}$$, and $$\theta$$ is in quadrant I, find $$\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.
13. If $$\tan \left(\theta \right)=\dfrac{12}{5}$$, and $$0\le \theta <\dfrac{\pi }{2}$$, find $$\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)$$.
14. If $$\tan \left(\theta \right)=4$$, and $$0\le \theta <\dfrac{\pi }{2}$$, find $$\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)$$.
15. Use a calculator to find sine, cosine, and tangent of the following values:
a. 0.15
b. 4
c. 70$$\mathrm{{}^\circ}$$
d. 283$$\mathrm{{}^\circ}$$
16. Use a calculator to find sine, cosine, and tangent of the following values:
a. 0.5
b. 5.2
c. 10$$\mathrm{{}^\circ}$$
d. 195$$\mathrm{{}^\circ}$$

Simplify each of the following to an expression involving a single trig function with no fractions.

17. $$\csc (t)\tan \left(t\right)$$

18. $$\cos (t)\csc \left(t\right)$$

19. $$\dfrac{\sec \left(t\right)}{\csc \left(t\right)\; }$$

20. $$\dfrac{\cot \left(t\right)}{\csc \left(t\right)}$$

21. $$\dfrac{\sec \left(t\right)-\cos \left(t\right)}{\sin \left(t\right)}$$

22. $$\dfrac{\tan \left(t\right)}{\sec \left(t\right)-\cos \left(t\right)}$$

23. $$\dfrac{1+\cot \left(t\right)}{1+\tan \left(t\right)}$$

24. $$\dfrac{1+\sin \left(t\right)}{1+\csc \left(t\right)}$$

25. $$\dfrac{\sin ^{2} \left(t\right)+\cos ^{2} \left(t\right)}{\cos ^{2} \left(t\right)}$$

26. $$\dfrac{1-\sin ^{2} \left(t\right)}{\sin ^{2} \left(t\right)}$$

Prove the identities.

27. $$\dfrac{\sin ^{2} \left(\theta \right)}{1+\cos \left(\theta \right)} =1-\cos \left(\theta \right)$$

28. $$\text{tan}^{2} (t) = \dfrac{1}{\text{cos}^(t)} - 1$$

29. $$\text{sec}(a) - \text{cos}(a) = \text{sin}(a) \text{tan}(a)$$

30. $$\dfrac{1 + \text{tan}^2(b)}{\text{tan}^2(b)} = \text{csc}^2(b)$$

31. $$\dfrac{\text{csc}^2 (x) - \text{sin}^2 (x)}{\text{csc} (x) + \text{sin} (x)} = \text{cos} (x) \text{cot} (x)$$

32. $$\dfrac{\text{sin} (\theta) - \text{cos} (\theta)}{\text{sec}(\theta) - \text{csc} (\theta)} = \text{sin} (\theta) \text{cos} (\theta)$$

33. $$\dfrac{\text{csc}^2 (\alpha) - 1}{\text{csc}^2 (\alpha) - \text{csc} (\alpha)} = 1 + \text{sin} (\alpha)$$

34. $$1 + \text{cot} (x) = \text{cos} (x) (\text{sec}(x) + \text{csc} (x))$$

35. $$\dfrac{1 + \text{cos} (u)}{\text{sin} (u)} = \dfrac{\text{sin} (u)}{1 - \text{cos}(u)}$$

36. $$2 \text{sec}^2 (t) = \dfrac{1 - \text{sin}(t)}{\text{cos}^2 (t)} + \dfrac{1}{1 - s \text{sin} (t)}$$

37. $$\dfrac{\text{sin}^4 (\gamma) - \text{cos}^4 (\gamma)}{\text{sin} (\gamma) - \text{cos} (\gamma)} = \text{sin} (\gamma) + \text{cos} (\gamma)$$

38. $$\dfrac{(1 + \text{cos}(A))(1 - \text{cos} (A))}{\text{sin} (A)} = \text{sin} (A)$$

1. $$\text{sec} (\theta) = \sqrt{2}$$, $$\text{csc} (\theta) = \sqrt{2}$$, $$\text{tan} (\theta) = 1$$, $$\text{cot} (\theta) = 1$$

3. $$\text{sec} (\theta) = -\dfrac{2\sqrt{3}}{3}$$, $$\text{csc} (\theta) = 2$$, $$\text{tan} (\theta) = -\dfrac{\sqrt{3}}{3}$$, $$\text{cot} (\theta) = -\sqrt{3}$$

5. $$\text{sec} (\theta) = -2$$, $$\text{csc} (\theta) = \dfrac{2\sqrt{3}}{3}$$, $$\text{tan} (\theta) = -\sqrt{3}$$, $$\text{cot} (\theta) = -\dfrac{\sqrt{3}}{3}$$

7. a. $$\text{sec} (135^{\circ}) = -\sqrt{2}$$
b. $$\text{csc} (210^{\circ}) = -2$$
c. $$\text{tan} (60^{\circ}) = \sqrt{3}$$
d. $$\text{cot} (225^{\circ}) = 1$$

9. $$\cos(\theta) = -\dfrac{\sqrt{7}}{4}$$, $$\sec (\theta) = -\dfrac{4\sqrt{7}}{7}$$, $$\csc(\theta) = \dfrac{4}{3}$$, $$\tan(\theta) = -\dfrac{3\sqrt{7}}{7}$$, $$\cot(\theta) = -\dfrac{\sqrt{7}}{3}$$

11. $$\sin(\theta) = -\dfrac{2\sqrt{2}}{3}$$, $$\csc(\theta) = -\dfrac{3\sqrt{2}}{3}$$, $$\sec(\theta) = -3$$, $$\tan(\theta) = 2\sqrt{2}$$, $$\cot(\theta) = \dfrac{\sqrt{2}}{4}$$

13. $$\sin(\theta) = \dfrac{12}{13}$$, $$\cos(\theta) = \dfrac{5}{13}$$, $$\sec(\theta) = \dfrac{13}{5}$$, $$\csc(\theta) = \dfrac{13}{12}$$, $$\cot(\theta) = \dfrac{5}{12}$$

15. a. sin(0.15) = 0.1494 cos(0.15) = 0.9888 tan(0.15) = 0.1511
b. sin(4) = -0.7568 cos(4) = -0.6536 tan(4) = 1.1578
c. sin($$70^{\circ}$$) = 0.9397 cos($$70^{\circ}$$) = 0.3420 tan($$70^{\circ}$$) = 2.7475
d. sin($$283^{\circ}$$) = -0.9744 cos($$283^{\circ}$$) = 0.2250 tan($$283^{\circ}$$) = -4.3315

17. sec($$t$$)

19. tan($$t$$)

21. tan($$t$$)

23. cot($$t$$)

25. $$(\sec(t))^2$$