Home fsc 1st year math solutions Double and Triple angle Identities Exercise 10.3

Double and Triple angle Identities Exercise 10.3

Trigonometric Identities 10.3
Trigonometric Identities 10.3
  1. Find the values of \sin 2 \alpha, \cos 2 \alpha and \tan 2 \alpha, when:
    i) \sin \alpha=\frac{12}{13}
    ii) \cos \alpha=\frac{3}{5}, where 0<\alpha<\frac{\pi}{2}
    Prove the following identities:
  2. \cot \alpha-\tan \alpha=2 \cot 2 \alpha
  3. \frac{\sin 2 \alpha}{1+\cos 2 \alpha}=\tan \alpha
  4. \frac{1-\cos \alpha}{\sin \alpha}=\tan \frac{\alpha}{2}
  5. \frac{\cos \alpha-\sin \alpha}{\cos \alpha+\sin \alpha}=\sec 2 \alpha-\tan 2 \alpha
  6. \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{\sin \frac{\alpha}{2}+\cos \frac{\alpha}{2}}{\sin \frac{\alpha}{2}-\cos \frac{\alpha}{2}}
  7. \frac{\operatorname{coses} \theta+\operatorname{coses} 2 \theta}{\sec \theta}=\cot \frac{\theta}{2}
  8. 1+\tan \alpha \tan 2 \alpha=\sec 2 \alpha
  9. \frac{2 \sin \theta \sin 2 \theta}{\cos \theta+\cos 3 \theta}=\tan 2 \theta \tan \theta
  10. \frac{\sin 3 \theta}{\sin \theta}-\frac{\cos 3 \theta}{\cos \theta}=2
  11. \frac{\cos 3 \theta}{\cos \theta}+\frac{\sin 3 \theta}{\sin \theta}=4 \cos 2 \theta
  12. \frac{\tan \frac{\theta}{2}+\cot \frac{\theta}{2}}{\cot \frac{\theta}{2}-\tan \frac{\theta}{2}}=\sec \theta
  13. \frac{\sin 3 \theta}{\cos \theta}+\frac{\cos 3 \theta}{\sin \theta}=2 \cot 2 \theta
  14. Reduce \sin ^4 \theta to an expression involving only function of multiples of \theta, raised to the first power
  15. Find the values of \sin \theta and \cos \theta without using table or calculator, when \theta is
    i) 18
    ii) 36
    iii) 54
    iv) 72^{\cric}
    Hence prove that: \cos 36 \cos 72 \cos 108 \cos 144^{\prime}=\frac{1}{16}
Previous articleApplications of basic Identities Exercise 10.2
Next articleSum, Difference and Product of Sines and Cosines Exercise 10.4
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