Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 134801 by deleteduser12 last updated on 07/Mar/21

Answered by EDWIN88 last updated on 07/Mar/21

$$13\theta =\pi \Rightarrow \cos 13\theta =-1$$$$\mathrm{let}:\mathrm{z}=\cos \theta \cos 2\theta \cos 3\theta \cos 4\theta \cos 5\theta \cos 6\theta$$$$2\mathrm{z}\sin \theta =\sin 2\theta \cos 2\theta \cos 3\theta \cos 4\theta \cos 5\theta \cos 6\theta$$$$4\mathrm{z}\sin \theta =\sin 4\theta \cos 3\theta \cos 4\theta \cos 5\theta \cos 6\theta$$$$8\mathrm{z}\sin \theta =\sin 8\theta \cos 3\theta \cos 5\theta \cos 6\theta$$$$\mathrm{note}\sin 8\theta =\sin \left(\frac{8\pi }{13}\right)=\sin (\pi -\frac{5\pi }{13})=\sin 5\theta$$$$8\mathrm{z}\sin \theta =\sin 5\theta \cos 3\theta \cos 5\theta \cos 6\theta$$$$16\mathrm{z}\sin \theta =\sin 10\theta \cos 3\theta \cos 6\theta$$$$\mathrm{note}\sin 10\theta =\sin \left(\frac{10\pi }{13}\right)=\sin (\pi -\frac{3\pi }{13})=\sin 3\theta$$$$16\mathrm{z}\sin \theta =\sin 3\theta \cos 3\theta \cos 6\theta$$$$32\mathrm{z}\sin \theta =\sin 6\theta \cos 6\theta$$$$64\mathrm{z}\sin \theta =\sin 12\theta \Rightarrow \mathrm{z}=\frac{\sin \left(\frac{12\pi }{13}\right)}{64\sin \left(\frac{\pi }{13}\right)}=\frac{1}{64}=\frac{1}{2^6}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com