Question

use the cofunction identities to find an angle θ between (0^{circ}) and (90^{circ}) that makes the statement true.
(cos\theta=sin(4\theta + 45^{circ}))
(\theta=square^{circ}) (type an integer or a simplified fraction.)

Explanation:

Step1: Use co - function identity

The co - function identity is \(\cos\theta=\sin(90^{\circ}-\theta)\).
So, if \(\cos\theta = \sin(4\theta + 45^{\circ})\), then \(90^{\circ}-\theta=4\theta + 45^{\circ}\).

Step2: Solve the equation for \(\theta\)

Add \(\theta\) to both sides of the equation \(90^{\circ}-\theta=4\theta + 45^{\circ}\):
\(90^{\circ}=4\theta+\theta + 45^{\circ}\), which simplifies to \(90^{\circ}=5\theta + 45^{\circ}\).
Subtract \(45^{\circ}\) from both sides: \(90^{\circ}-45^{\circ}=5\theta\), so \(45^{\circ}=5\theta\).
Divide both sides by \(5\): \(\theta=\frac{45^{\circ}}{5}=9^{\circ}\).

Answer:

\(9\)