Solve the equation for $\theta$ $( 0 \leq \theta \leq 2 \pi )$ . For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. $\cos 2 \theta + 3 \cos \theta + 2 = 0$
A) $\frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \pi$
B) $\frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , 2 \pi$
C) $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \pi$
D) $0 , \frac { \pi } { 3 } , \frac { 4 \pi } { 3 }$
E) $0 , \frac { \pi } { 3 } , \frac { 4 \pi } { 3 } , \pi$

Question

Quizplus · Accepted Answer

To solve the equation $\cos 2\theta + 3\cos \theta + 2 = 0$, we can use the double angle identity for cosine, which is $\cos 2\theta = 2\cos^2 \theta - 1$. Substituting this into the equation gives us $2\cos^2 \theta - 1 + 3\cos \theta + 2 = 0$, which simplifies to $2\cos^2 \theta + 3\cos \theta + 1 = 0$. Factoring this quadratic equation in terms of $\cos \theta$ yields $(2\cos \theta + 1)(\cos \theta + 1) = 0$, leading to $\cos \theta = -\frac{1}{2}$ or $\cos \theta = -1$. For $\cos \theta = -\frac{1}{2}$, the solutions within $0 \leq \theta \leq 2\pi$ are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. For $\cos \theta = -1$, the solution within $0 \leq \theta \leq 2\pi$ is $\theta = \pi$. Therefore, the correct answer is $\frac{2\pi}{3}, \frac{4\pi}{3}, \pi$, which corresponds to choice A.

Solve the Equation For θ\thetaθ (0≤θ≤2π)( 0 \leq \theta \leq 2 \pi )(0≤θ≤2π)

Solve the Equation For $\theta$ $( 0 \leq \theta \leq 2 \pi )$