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Homefsc 1st year math solutionsSum, Difference and Product of Sines and Cosines Exercise 10.4

Sum, Difference and Product of Sines and Cosines Exercise 10.4

  1. Express the following products as sums or differences:
    i) 2 \sin 3 \theta \cos \theta
    ii) 2 \cos 5 \theta \sin 3 \theta
    iii) \sin 5 \theta \cos 2 \theta
    iv) 2 \sin 7 \theta \sin 2 \theta
    v) \cos (x+y) \sin (x-y)
    vi) \cos (2 x+30) \cos \left(2 x-30^{\circ}\right)
    vii) \sin 12^{\circ} \sin 46^{\prime}
    viii) \sin \left(x+45^{\circ}\right) \sin \left(x-45^7\right)
  2. Express the following sums or differences as products:
    i) \sin 5 \theta+\sin 3 \theta
    ii) \sin 8 \theta-\sin 4 \theta
    iii) \cos 6 \theta+\cos 3 \theta
    iv) \cos 7 \theta-\cos \theta
    v) \cos 12^{\circ}+\cos 48^{\circ}
    vi) \sin \left(x+30^{\circ}\right)+\sin \left(x-30^{\circ}\right)
  3. Prove the following identities:
    i) \frac{\sin 3 x-\sin x}{\cos x-\cos 3 x}=\cot 2 x
    ii) \frac{\sin 8 x+\sin 2 x}{\cos 8 x+\cos 2 x}=\tan 5 x
    iii) \frac{\sin \alpha-\sin \beta}{\sin \alpha+\sin \beta}=\tan \frac{\alpha-\beta}{2} \cot \frac{\alpha+\beta}{2}
  4. Prove that:
    i) \cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0
    ii) \sin \left(\frac{\pi}{4}-\theta\right) \sin \left(\frac{\pi}{4}+\theta\right)=\frac{1}{2} \cos 2 \theta iii)\frac{\sin \theta+\sin 3 \theta+\sin 5 \theta+\sin 7 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta+\cos 7 \theta}=\tan 4 \theta
  5. Prove that:
    i) \cos 20^{\circ} \cos 40^{\prime} \cos 60^{\circ} \cos 80^{\circ}=\frac{1}{16}
    ii) \sin \frac{\pi}{9} \sin \frac{2 \pi}{9} \sin \frac{\pi}{3} \sin \frac{4 \pi}{9}=\frac{3}{16}
    iii) \sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\prime}=\frac{1}{16}
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