# Find all exact angles,x in the interval[-pi, pi] that satisfy (a)2cos x=√3 (b)√3 sin x = cos x (c) 4sin²x = √(12) sin x (d) sec² x=2 ?

Dec 14, 2017

a. $x = \pm \frac{\pi}{6}$
b. $x = \frac{\pi}{6} , \mathmr{and} x = \frac{7 \pi}{6}$
c. $x = \frac{\pi}{3} , \mathmr{and} x = \frac{2 \pi}{3}$
d. ${63}^{\circ} 43 , \mathmr{and} x = {243}^{\circ} 43$

#### Explanation:

(a). $2 \cos x = \sqrt{3}$ --> $\cos x = \frac{\sqrt{3}}{2}$
Trig table and unit circle gives 2 solutions:
$x = \pm \frac{\pi}{6}$, or $x = \pm {30}^{\circ}$
(b). $\sqrt{3} \sin x = \cos x$
Divide both sides by cos x (condition $\cos x \ne 0$)
$\sqrt{3} \tan x = 1$--> $\tan x = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
Trig table and unit circle give 2 solutions:
$x = \frac{\pi}{6}$ and $x = \frac{\pi}{6} + \pi = \frac{7 \pi}{6}$
(c). $4 {\sin}^{2} x = s q r 12 \sin x = 2 \sqrt{3} \sin x$.
Simplify by sin x (condition $\sin x \ne 0$).
$2 \sin x = \sqrt{3}$ --> $\sin x = \frac{\sqrt{3}}{2}$
Trig table and unit circle -->
$x = \frac{\pi}{3}$, and $x = \frac{2 \pi}{3}$
(d). $\sin x . \sec x = 2$.
$\left(\sin x\right) \left(\frac{1}{\cos} x\right) = \tan x = 2$
Calculator and unit circle give 2 solutions:
$x = {63}^{\circ} 43$, and $x = 63.43 + 180 = {243}^{\circ} 43$