If $\cos (90°-\theta )=\sin \left(2\theta \right)$, what is $\theta$?

The correct answer is 60°.

To find the value of $\theta$, we start with the given trigonometric equation:

$\cos (90°-\theta )=\sin \left(2\theta \right)$

Using the co-function identity of trigonometry, we know that:

$\cos (90°-\theta )=\sin \left(\theta \right)$

Substituting this co-function identity back into the given equation, we get:

$\sin \left(\theta \right)=\sin \left(2\theta \right)$

Recall the double-angle formula for sine, which states that $\sin \left(2\theta \right)=2\sin \left(\theta \right)\cos \left(\theta \right)$. Replacing $\sin \left(2\theta \right)$ with this identity gives:

$\sin \left(\theta \right)=2\sin \left(\theta \right)\cos \left(\theta \right)$

Subtracting $\sin \left(\theta \right)$ from both sides of the equation gives:

$2\sin \left(\theta \right)\cos \left(\theta \right)-\sin \left(\theta \right)=0$

Factoring out $\sin \left(\theta \right)$ yields:

$\sin \left(\theta \right)\left(2\cos \right(\theta )-1)=0$

This equation gives two possible scenarios:

1. $\sin \left(\theta \right)=0$, which implies $\theta =0°$ (or multiples of 180°).

2. $2\cos \left(\theta \right)-1=0$

Solving the second case for $\cos \left(\theta \right)$:

$2\cos \left(\theta \right)=1$

$\cos \left(\theta \right)=\frac{1}{2}$

For an acute angle, the value of $\theta$ where $\cos \left(\theta \right)=\frac{1}{2}$ is:

$\theta =60°$

Thus, $\theta =60°$ matches the given correct option.