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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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Apkplot.com is an all-in-one educational site that has answers to math problems and free past papers and MCQs to help people prepare for the MDCAT and ECAT. The website is based on FSC and covers the subjects of math, physics, chemistry, and biology for Parts 01 and 02. Students can use the website's large database of answers to math problems and step-by-step explanations to help them. Students taking FSC Parts 01 and 02, as well as the MCAT and ECAT, can also find past papers on the website. These papers help students understand the format of the exam and prepare well for the real test. The MCQs are also a great way for students to test their knowledge and figure out what they need to learn more about.
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