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  • Find the value of  <img alt="\cos 20^\circ \cos 40^\circ \cos 80^\circ" src="https://learn.careers360.com/latex-image/?%5Ccos%2020%5E%5Ccirc%20%5Ccos%2040%5E%5Ccirc%20%5Ccos%2080%5E%5Ccir

Find the value of  \cos 20^\circ \cos 40^\circ \cos 80^\circ

Option: 1

\frac{1}{2}


Option: 2

\frac{1}{4}


Option: 3

\frac{1}{8}


Option: 4

\frac{\sqrt{3}}{8}


Answers (1)

best_answer

Triple Angle Formula -

Triple Angle Formula

 

\\\mathrm{1.\;\;\sin 3A=3\;sin A-4\;\sin^3A}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=4\;\sin (60^{\circ}-A)\cdot\sin A\cdot\sin (60^{\circ}+A)}\\\\\mathrm{2.\;\;\cos 3A= 4\;\cos^3A}-3\;\cos A\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=4\;\cos (60^{\circ}-A)\cdot\cos A\cdot\cos (60^{\circ}+A)}

 

\\\mathrm{3.\;\;\tan 3A=\frac{3\tan A-\tan^3A}{1-3\tan^2 A}}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;\tan (60^{\circ}-A)\cdot\tan A\cdot\tan (60^{\circ}-A)}

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\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}= \cos (60^{\circ}-20^{\circ})\cdot\cos 20^{\circ}\cdot\cos (60^{\circ}+20^{\circ})\\= \frac{1}{4}\cos(60^{\circ}) \ \ \ \because \cos 3A=4 \cos (60^{\circ}-A)\cdot\cos A\cdot\cos (60^{\circ}+A)\\ =\frac{1}{8}

Posted by

Rishabh

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