if tg(0,5x) = −2 find ((sin(x) + 2)/(cos(x)


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Question Number 148987 by mathdanisur last updated on 02/Aug/21

$i f t g (0, 5 x) = - 2$ $f i n d \frac{s i n (x) + 2}{c o s (x) - 3} = ?$
	Answered by EDWIN88 last updated on 02/Aug/21
	$tan ((1/2)x)=−2 → { (((1/2)x in 2^(nd) quadrant)),(((1/2)x in 4^(th) quadrant)) :} ⇒((sin x+2)/(cos x−3)) = ((2sin (1/2)x cos (1/2)x+2)/(2cos^2 (1/2)x−4)) = ((2((2/( (√5))))(−(1/( (√5))))+2)/(2(−(1/( (√5))))^2 −4)) =(((−4+10)/5)/((2−20)/5)) = (6/(−18))=−(1/3)$
	$\tan (\frac{1}{2} x) = - 2 \to {\begin{cases} \frac{1}{2} x i n 2^{n d} q u a d r a n t \\ \frac{1}{2} x i n 4^{t h} q u a d r a n t \end{cases}$ $\Rightarrow \frac{\sin x + 2}{\cos x - 3} = \frac{2 \sin \frac{1}{2} x \cos \frac{1}{2} x + 2}{2 \cos^{2} \frac{1}{2} x - 4}$ $= \frac{2 (\frac{2}{\sqrt{5}}) (- \frac{1}{\sqrt{5}}) + 2}{2 {(- \frac{1}{\sqrt{5}})}^{2} - 4}$ $= \frac{\frac{- 4 + 10}{5}}{\frac{2 - 20}{5}} = \frac{6}{- 18} = - \frac{1}{3}$
	Commented by mathdanisur last updated on 02/Aug/21

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