
# 5.4E: The Other Trigonometric Functions (Exercises)

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## 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 $$\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)$$.

1. Use a calculator to find sine, cosine, and tangent of the following values:

a. 0.15 b. 4 c. 70$$\mathrm{{}^\circ}$$ d. 283$$\mathrm{{}^\circ}$$

1. 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.

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

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

3. $$\frac{\sec \left(t\right)}{\csc \left(t\right)\; }$$

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

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

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

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

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

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

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

Prove the identities.

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

Section 5.5 Right Triangle Trigonometry 391