# How do you solve 5sin^2x-8sinx=3 in the interval [0,360]?

May 1, 2018

90^@; 216^@87; 323^@13

#### Explanation:

$f \left(x\right) = 5 {\sin}^{2} x - 8 \sin x - 3 = 0$
Solve the above quadratic equation for sin x.
Since a + b + c = 0, the shortcut gives:
sin x = 1 and $\sin x = \frac{c}{a} = - \frac{3}{5}$
a. sin x = 1 --> $x = \frac{\pi}{2} = {90}^{\circ}$
b. $\sin = - \frac{3}{5}$
Calculator and unit circle give 2 solutions:
$x = - {36}^{\circ} 87$, or $x = 360 - 36.87 = {323}^{\circ} 13$ (co-terminal), and:
$x = 180 - \left(- 36.87\right) = {216}^{\circ} 87$