$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 5.4E: The Other Trigonometric Functions (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Section 5.4 Exercises

1. If $$\theta =\frac{\pi \; }{4}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.

2. If $$\theta =\frac{7\pi \; }{4}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.

3. If $$\theta =\frac{5\pi \; }{6}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.

4. If $$\theta =\frac{\pi \; }{6}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)$$.

5. If $$\theta =\frac{2\pi \; }{3}$$ , find exact values for $$\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;$$.

6. If $$\theta =\frac{4\pi \; }{3}$$ , 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 $$\sin \left(\theta \right)=\frac{3}{4}$$, 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 $$\sin \left(\theta \right)=\frac{2}{7}$$, 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 $$\cos \left(\theta \right)=-\frac{1}{3}$$, 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 $$\cos \left(\theta \right)=\frac{1}{5}$$, 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 $$\tan \left(\theta \right)=\frac{12}{5}$$, and $$0\le \theta <\frac{\pi }{2}$$, 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 $$0\le \theta <\frac{\pi }{2}$$, 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. $$\frac{\sec \left(t\right)}{\csc \left(t\right)\; }$$

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

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

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

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

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

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

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

Prove the identities.

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

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