Question

If $0^{\circ }<x<{90}^{\circ }$ and $2\sin x+15{\cos }^{2}x=7,$ then what is the value of $\tan x?$

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $\frac{4}{3}$
• D. $\frac{3}{4}$

Answer 1

The correct option is C $\frac{4}{3}$

$2\sin x+15{\cos }^{2}x=7$

$2\sin x+15(1-{\sin }^{2}x)-7=0$

$2\sin x-15{\sin }^{2}x+8=0$

$15{\sin }^{2}x-2\sin x-8=0$

$\sin x=\frac{2\pm \sqrt{4+\mathrm{4.15.8}}}{2.15}$

$=\frac{2\pm \sqrt{4+480}}{30}$

$=\frac{2\pm 22}{30}$

$=\frac{24}{30}$, $\frac{-20}{30}$

$\sin x>0$ $(\because 0<x<{90}^o)$

$\sin x=\frac{24}{30}=\frac{4}{5}$

$\cos x=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5}$

$\tan x=\frac{\sin x}{\cos x}=\frac{4/5}{3/5}=\frac{4}{3}$

$\therefore \tan x=\frac{4}{3}$.