Question

solve the following trigonometric equation for $0 \leq \theta < 2\pi$.
$2 \cos 3\theta + 1 = 0$
how many solutions to the equation are found on the interval?
6
two of the solutions are $2\pi/9$ and select .
select
$10\pi/9$
$11\pi/18$
$11\pi/9$
$7\pi/9$
question 15

Explanation:

Step1: Isolate the cosine term

$2\cos3\theta = -1$
$\cos3\theta = -\frac{1}{2}$

Step2: Find general angle for cosine

The general solution for $\cos x = -\frac{1}{2}$ is $x = \frac{2\pi}{3} + 2\pi k$ or $x = \frac{4\pi}{3} + 2\pi k$, where $k$ is any integer. Substitute $x=3\theta$:
$3\theta = \frac{2\pi}{3} + 2\pi k$ or $3\theta = \frac{4\pi}{3} + 2\pi k$

Step3: Solve for $\theta$

$\theta = \frac{2\pi}{9} + \frac{2\pi k}{3}$ or $\theta = \frac{4\pi}{9} + \frac{2\pi k}{3}$

Step4: Find valid $\theta$ in $0\leq\theta<2\pi$

• For $\theta = \frac{2\pi}{9} + \frac{2\pi k}{3}$:

$k=0$: $\theta=\frac{2\pi}{9}$; $k=1$: $\theta=\frac{2\pi}{9}+\frac{6\pi}{9}=\frac{8\pi}{9}$; $k=2$: $\theta=\frac{2\pi}{9}+\frac{12\pi}{9}=\frac{14\pi}{9}$

• For $\theta = \frac{4\pi}{9} + \frac{2\pi k}{3}$:

$k=0$: $\theta=\frac{4\pi}{9}$; $k=1$: $\theta=\frac{4\pi}{9}+\frac{6\pi}{9}=\frac{10\pi}{9}$; $k=2$: $\theta=\frac{4\pi}{9}+\frac{12\pi}{9}=\frac{16\pi}{9}$

Step5: Match with given options

From the valid solutions, $\frac{10\pi}{9}$ is in the selectable options.

Answer:

Number of solutions: 6
The missing solution: 10π/9