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Given the equation:

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

We can use the double-angle identity for sine:

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

The equation becomes:

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

Dividing both sides by $\cos(\theta)$ (assuming $\cos(\theta) \neq 0$):

$1 = 2\sin(\theta)$

Solving for $\sin(\theta)$:

$\sin(\theta) = \frac{1}{2}$

The values of $\theta$ in the interval [0, 360) where $\sin(\theta) = \frac{1}{2}$ are $\theta = 30^\circ$ and $\theta = 150^\circ$.

However, we also need to consider the case where $\cos(\theta) = 0$:

$\cos(\theta) = 0$ for $\theta = 90^\circ$ and $\theta = 270^\circ$.

Therefore, the angles that satisfy the equation are: $30^\circ$, $90^\circ$, $150^\circ$, and $270^\circ$.

Given:

$cos( heta) = sin(2 heta)$

Using $sin(2 heta) = 2sin( heta)cos( heta)$, we get:

$cos( heta) = 2sin( heta)cos( heta)$

Dividing by $cos( heta)$:

$1 = 2sin( heta)$

$sin( heta) = frac{1}{2}$

Possible $ heta$: $30^circ$, $150^circ$.

For $cos( heta) = 0$: $ heta = 90^circ$, $270^circ$.

Hence, solutions: $30^circ$, $90^circ$, $150^circ$, $270^circ$.

From $cos( heta) = sin(2 heta)$ and $sin(2 heta) = 2sin( heta)cos( heta)$:

$cos( heta) = 2sin( heta)cos( heta)$

Dividing by $cos( heta)$:

$sin( heta) = frac{1}{2}$

Values: $30^circ$, $150^circ$.

Also $cos( heta) = 0$: $90^circ$, $270^circ$.

Final $ heta$: $30^circ$, $90^circ$, $150^circ$, $270^circ$.

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