# Number of solutions of sin^2θ+3cosθ=3 in [-π,π] ?

Jun 23, 2018

One answer: t = 0

#### Explanation:

Replace in the equation ${\sin}^{2} t$ by $\left(1 - {\cos}^{2} t\right)$:, then, solve the quadratic equation for cos t:
$1 - {\cos}^{2} t + 3 \cos t - 3 = 0$
$- {\cos}^{2} t + 3 \cos t - 2 = 0$
Sine a + b + c = 0, use shortcut. The 2 real roots are:
cos x = 1 and $\cos x = \frac{c}{a} = 2$ (rejected as > 1)
cos x = 1.
Unit circle gives --> t = 0, and $t = 2 \pi$
Inside the interval $\left(- \pi , \pi\right)$, there is only one answer:
t = 0
Check
t = 0 --> sin^2 t = 0 --> 3cos t = 3
sin^2 t + 3cos t = 0 + 3 = 3. Proved.