Find the degree measure of the least positive angle $\theta$ for which
\[\cos 5^\circ = \sin 25^\circ + \sin \theta.\]
From the given equation,
\begin{align*}
\sin \theta &= \cos 5^\circ - \sin 25^\circ \\
&= \cos 5^\circ - \cos 65^\circ.
\end{align*}By the sum-to-product formula,
\begin{align*}
\cos 5^\circ - \cos 65^\circ &= -2 \sin 35^\circ \sin (-30^\circ) \\
&= \sin 35^\circ.
\end{align*}Thus, the smallest such $\theta$ is $\boxed{35^\circ}.$
35
Level 3
Precalculus
Problem: Find the degree measure of the least positive angle $\theta$ for which
\[\cos 5^\circ = \sin 25^\circ + \sin \theta.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation \(\cos 5^\circ = \sin 25^\circ + \sin \theta\), we first need to express \(\sin 25^\circ\) in terms of cosine. Using the identity \(\sin x = \cos(90^\circ - x)\), we find that \(\sin 25^\circ = \cos 65^\circ\).

Next, we rewrite the equation as:
\[
\cos 5^\circ = \cos 65^\circ + \sin \theta.
\]

To isolate \(\sin \theta\), we rearrange the equation:
\[
\sin \theta = \cos 5^\circ - \cos 65^\circ.
\]

Now, we apply the cosine subtraction formula:
\[
\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right).
\]
Setting \(A = 5^\circ\) and \(B = 65^\circ\), we calculate:
\[
\frac{A+B}{2} = \frac{5^\circ + 65^\circ}{2} = 35^\circ,
\]
\[
\frac{A-B}{2} = \frac{5^\circ - 65^\circ}{2} = -30^\circ.
\]
Thus, we have:
\[
\cos 5^\circ - \cos 65^\circ = -2 \sin(35^\circ) \sin(-30^\circ).
\]
Since \(\sin(-30^\circ) = -\frac{1}{2}\), we substitute:
\[
\cos 5^\circ - \cos 65^\circ = -2 \sin(35^\circ) \left(-\frac{1}{2}\right) = \sin(35^\circ).
\]

This leads us to:
\[
\sin \theta = \sin(35^\circ).
\]

The solutions for \(\sin \theta = \sin(35^\circ)\) are given by:
\[
\theta = 35^\circ + k \cdot 360^\circ \quad \text{or} \quad \theta = 180^\circ - 35^\circ + k \cdot 360^\circ,
\]
for any integer \(k\). The second solution simplifies to:
\[
\theta = 145^\circ + k \cdot 360^\circ.
\]

Since we are looking for the least positive angle, we take \(k = 0\) for both cases. The least positive angle is \(35^\circ\), as \(145^\circ\) is greater than \(35^\circ\).

Thus, the final answer is \(35\).<|END_OF_LATENT|> The final answer is $35$.