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## 30 Trigonemetric Problems and Equations $1. \; \sqrt{ 1 + \sin x } + \sqrt{ 1 - \sin x} = 1 + \cos x \\ 2. \; \sin \alpha = \dfrac{1}{4} \; and \; \alpha \in [ 0 ; \pi / 4 ] . \; Find \; \tan 2 \alpha \\ 3. \; \tan 5x = \sin^2 x \cdot \tan 5x \\ 4. \; \sin x + \sin 2x + \sin 3x = 1 + \cos x + \cos 2x \\ 5. \; \sin x \cdot \sin 3x + \sin 4x \cdot \sin 8x = 0 \\ 6. \; \sin 3x + \sin 5x = \sin 4x \\ 7. \; \cos 3x \cdot \cos x - \cos 5x \cdot \cos 7x = 0 \\ 8. \; \sin 3x \cdot \sin 5x + 2 \sin^2 x = 1 \\ 9. \; \sin 2x \cdot \sin 6x = \cos x \cdot \cos 3x \\ 10. \; \cos^4 x + \sin^4 x = \sin 2x - 0.5 \\ 11. \; \cos 2x = \cos x - \sin x \\ 12. \; \sin^6 x + \cos^6 x = \dfrac{1}{4} \cdot \sin^2 2x \\ 13. \; \cos 7x + \sin 8x = \cos 3x - \sin 2x \\ 14. \; 1 + \cos x + \cos 2x = 0 \\ 15. \; \cos 2x - 5 \sin x - 3 = 0 \\ 16. \; 4 \cdot \sin x \cdot \sin 2x \cdot \sin 3x = \sin 4x \\ 17. \; \sin 2x = \cos^4 \dfrac{x}{2} - \sin^4 \frac{x}{2} \\ 18. \; 2 \cos x + \sin x + 2 = 0 \\ 19. \; \sin x + \sin 3x = 4 \cos^3 x \\ 20. \; \sin^3 x \cdot \cos x - \cos^3 x \cdot \sin x = \dfrac{ \sqrt{2} }{8} \\ 21. \; 1 - \sin 2x = \cos x - \sin x \\ 22. \; 3 \cdot \tan 3x - 4 \cdot \tan 2x = \tan^2 2x \cdot \tan 3x \\ 23. \; 3 \cdot ( 1 - \sin x ) = 1 + \cos 2x \\ 24. \; \tan \dfrac{3x}{5} \cdot \cot \dfrac{5x}{3} = 1 - \sec \dfrac{3x}{5} \cdot \csc \dfrac{5x}{3} \\ 25. \; \sin ( x + 25^\circ ) \cdot \sin ( x - 20^\circ ) = \sin ( 70^\circ + x ) \cdot \sin ( 65^\circ - x ) \\ 26. \; \sin 2x + \cos 2x + \sin x + \cos x + 1 = 0 \\ 27. \; \sin 3x = \cos x - \sin x \\ 28. \; Find \; \sin ( 5 \arcsin x ) . \\ 29. \; \cos 7x + \sin^2 2x = \cos^2 2x - \cos x \\ 30. \; \sin x + \sin 2x + \sin 3x + \sin 4x = 0 \\$

Categories: Trigonometry
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