Question

Determine all solutions to the equation 1 over sine theta equals 2 times sine theta on the interval [0, 2π).

Answer 1

The solution to the trigonometric equation in the interval is given by:

$x = \frac{\pi}{4}$

What is the trigonometric equation?

In this problem, the equation is given by:

$\frac{1}{\sin{\theta}} = 2\cos{\theta}$

Applying cross multiplication, we have that:

$2\sin{\theta}\cos{\theta} = 1$

Applying a trigonometric identity, we have that:

$\sin{2x} = 1$

We have that on the interval [0, 2π), $\sin{\left(\frac{\pi}{2}\right)} = 1$

Hence:

$\sin{2x} = 1$

$\sin{2x} = \sin{\left(\frac{\pi}{2}\right)}$

$2x = \frac{\pi}{2}$

$x = \frac{\pi}{4}$

For $\frac{3\pi}{4}$, we have that:

$\sin{2x} = \sin{\left(\frac{3\pi}{2}\right)} = -1$

Hence only the first solution is correct.

More can be learned about trigonometric equations at https://brainly.com/question/24680641