5.4E: The Other Trigonometric Functions (Exercises)

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- 5.4: The Other Trigonometric Functions
- 5.5: Right Triangle Trigonometry

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Section 5.4 Exercise

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

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

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

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

Answer

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

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