Question

5cos theta+12sin theta= 13 find tan theta?​

Answer 1

Topic :-

Trigonometry

Given :-

$\sf{5\cos\theta+12\sin\theta=13}$

To Find :-

$\sf{\tan\theta=?}$

Solution :-

$\sf{\implies5\cos\theta+12\sin\theta=13}$

Divide whole equation with cosθ,

$\sf{\implies 5\cdot\dfrac{\cos\theta}{\cos\theta}+12\cdot\dfrac{\sin\theta}{\cos\theta}=\dfrac{13}{\cos\theta}}$

$\sf{\implies 5+12\tan\theta=13\sec\theta}$

Squaring both sides,

$\sf{\implies (5+12\tan\theta)^2=(13\sec\theta)^2}$

$\sf{\implies 25+2(5)(12\tan\theta)+144\tan^2\theta=169\sec^2\theta}$

$\sf{(\because (a+b)^2=a^2+2ab+b^2)}$

$\sf{\implies 25+120\tan\theta+144\tan^2\theta=169\sec^2\theta}$

$\sf{\implies 25+120\tan\theta+144\tan^2\theta=169(1+\tan^2\theta)}$

$(\because \sec^2\theta=1+\tan^2\theta)$

$\sf{\implies 25+120\tan\theta+144\tan^2\theta=169+169\tan^2\theta}$

$\sf{\implies 0=169\tan^2\theta-144\tan^2\theta-120\tan\theta+169-25}$

$\sf{\implies 25\tan^2\theta-120\tan\theta+144=0}$

Splitting middle term and factorising it,

$\sf{\implies 25\tan^2\theta-60\tan\theta-60\tan\theta+144=0}$

$\sf{\implies{ 5\tan \theta (5 \tan \theta-12)}}$

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