Question

solve the equation in degrees for all exact solutions where appropriate. round approximate answers in degrees to the nearest tenth. write answers using the least possible nonnegative angle measures.
2\cos^{2}2\theta=-1 - 3\cos2\theta
choose the correct solution set below.
a. \\{60^{\circ}+180^{\circ}n,120^{\circ}+180^{\circ}n, where n is any integer\\}
b. \\{120^{\circ}+180^{\circ}n,240^{\circ}+180^{\circ}n, where n is any integer\\}
c. \\{60^{\circ}+180^{\circ}n,90^{\circ}+180^{\circ}n,120^{\circ}+180^{\circ}n, where n is any integer\\}
d. \\{120^{\circ}+180^{\circ}n,180^{\circ}+180^{\circ}n,240^{\circ}+180^{\circ}n, where n is any integer\\}
e. \\{0^{\circ}+180^{\circ}n,30^{\circ}+180^{\circ}n,330^{\circ}+180^{\circ}n, where n is any integer\\}

Explanation:

Step1: Let \(x = 2\theta\)

The equation becomes \(2\cos^{2}x=-1 - 3\cos x\).

Step2: Rearrange the equation

\(2\cos^{2}x+3\cos x + 1 = 0\).

Step3: Factor the quadratic equation

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

Step4: Solve for \(t\)

\(2t+1 = 0\) gives \(t=-\frac{1}{2}\); \(t + 1=0\) gives \(t=-1\).

Step5: Solve for \(x\) when \(\cos x=-1\)

\(x=(180 + 360n)^{\circ}\), \(n\in Z\).

Step6: Solve for \(x\) when \(\cos x=-\frac{1}{2}\)

\(x=(120 + 360n)^{\circ}\) or \(x=(240+360n)^{\circ}\), \(n\in Z\).

Step7: Substitute back \(x = 2\theta\)

When \(x=(180 + 360n)^{\circ}\), \(\theta=(90 + 180n)^{\circ}\); when \(x=(120 + 360n)^{\circ}\), \(\theta=(60 + 180n)^{\circ}\); when \(x=(240+360n)^{\circ}\), \(\theta=(120 + 180n)^{\circ}\).

Answer:

B. \(\{120^{\circ}+180^{\circ}n,240^{\circ}+180^{\circ}n,\text{where }n\text{ is any integer}\}\)