which shows all the exact solutions of sin²x + 3cosx - 1 = 2? give your answer in radians. 0 2π 2kπ π + 2kπ

Explanation:

Step1: Use trig identity

Recall $\sin^{2}x = 1-\cos^{2}x$. Substitute into the equation: $1 - \cos^{2}x+3\cos x - 1=2$.

Step2: Simplify the equation

Simplify the left - hand side to get $-\cos^{2}x + 3\cos x=2$. Rearrange to $\cos^{2}x-3\cos x + 2 = 0$.

Step3: Factor the quadratic equation

Let $t=\cos x$, then $t^{2}-3t + 2=0$. Factoring gives $(t - 1)(t - 2)=0$.

Step4: Solve for t

Set each factor equal to zero: $t - 1=0$ gives $t = 1$ and $t - 2=0$ gives $t = 2$. But since $-1\leqslant\cos x\leqslant1$, we discard $t = 2$.

Step5: Solve for x

Since $\cos x=1$, then $x = 2k\pi$, where $k\in\mathbb{Z}$.

Answer:

$2k\pi$