
# 7.3E: Double Angle Identities (Exercises)

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## Section 7.3 Exercises

1. If $$\sin \left(x\right)=\frac{1}{8}$$ and x is in quadrant I, then find exact values for (without solving for x):

a. $$\sin \left(2x\right)$$ b. $$\cos \left(2x\right)$$ c. $$\tan \left(2x\right)$$

2. If $$\cos \left(x\right)=\frac{2}{3}$$ and x is in quadrant I, then find exact values for (without solving for x):

a. $$\sin \left(2x\right)$$ b. $$\cos \left(2x\right)$$ c. $$\tan \left(2x\right)$$

Simplify each expression. $3. \cos ^{2} \left(28{}^\circ \right)-\sin ^{2} (28{}^\circ ) 4. 2\cos ^{2} \left(37{}^\circ \right)-1$ $5. 1-2\sin ^{2} (17{}^\circ ) 6. \cos ^{2} \left(37{}^\circ \right)-\sin ^{2} (37{}^\circ )$ $7. \cos ^{2} \left(9x\right)-\sin ^{2} (9x) 8. \cos ^{2} \left(6x\right)-\sin ^{2} (6x)$ $9. 4\sin \left(8x\right){\rm cos}(8x) 10. 6\sin \left(5x\right){\rm cos}(5x)$

Solve for all solutions on the interval $$[0,\; 2\pi )$$. $11. 6\sin \left(2t\right)+9\sin \left(t\right)=0 12. 2\sin \left(2t\right)+3\cos \left(t\right)=0$ $13. 9\cos \left(2\theta \right)=9\cos ^{2} \left(\theta \right)-4 14. 8\cos \left(2\alpha \right)=8\cos ^{2} \left(\alpha \right)-1$ $15. \sin \left(2t\right)=\cos \left(t\right) 16. \cos \left(2t\right)=\sin \left(t\right)$ $17. \cos \left(6x\right)-\cos \left(3x\right)=0 18. \sin \left(4x\right)-\sin \left(2x\right)=0$

Use a double angle, half angle, or power reduction formula to rewrite without exponents. $19. \cos ^{2} (5x)~ 20. \cos ^{2} (6x)$ $21. \sin ^{4} (8x) 22. \sin ^{4} \left(3x\right)$ $23. \cos ^{2} x\sin ^{4} x 24. \cos ^{4} x\sin ^{2} x$

25. If $$\csc \left(x\right)=7$$ and $$90{}^\circ <x<180{}^\circ$$, then find exact values for (without solving for x):

1. $$\sin \left(\frac{x}{2} \right)$$ b. $$\cos \left(\frac{x}{2} \right)$$ c. $$\tan \left(\frac{x}{2} \right)$$

26. If $$\sec \left(x\right)=4$$ and $$270{}^\circ <x<360{}^\circ$$, then find exact values for (without solving for x):

1. $$\sin \left(\frac{x}{2} \right)$$ b. $$\cos \left(\frac{x}{2} \right)$$ c. $$\tan \left(\frac{x}{2} \right)$$

Prove the identity. $27. \left(\sin t-\cos t\right)^{2} =1-\sin \left(2t\right)$ $28. \left(\sin ^{2} x-1\right)^{2} =\cos \left(2x\right)+\sin ^{4} x$ $29. \sin \left(2x\right)=\frac{2\tan \left(x\right)}{1+\tan ^{2} \left(x\right)}$ $30. \tan \left(2x\right)=\frac{2\sin \left(x\right)\cos \left(x\right)}{2\cos ^{2} \left(x\right)-1}$ $31. \cot \left(x\right)-\tan \left(x\right)=2\cot \left(2x\right)$ $32. \frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)} =\tan \left(\theta \right)$ $33. \cos \left(2\alpha \right)=\frac{1-\tan ^{2} \left(\alpha \right)}{1+\tan ^{2} \left(\alpha \right)}$ $34. \frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos \left(t\right)} =\frac{2\cos \left(t\right)}{2\sin \left(t\right)-1}$ $35. \sin \left(3x\right)=3\sin \left(x\right)\cos ^{2} \left(x\right)-\sin ^{3} (x)$ $36. \cos \left(3x\right)=\cos ^{3} (x)-3\sin ^{2} (x)\cos \left(x\right)$

Section 7.4 Modeling Changing Amplitude and Midline 497