To solve the equation [tex]\(\sin(\theta + 10^\circ) = \cos(\theta)\)[/tex], we start with a key trigonometric identity.
We know that:
[tex]\[
\sin(x) = \cos(90^\circ - x)
\][/tex]
Therefore, we can rewrite [tex]\(\sin(\theta + 10^\circ)\)[/tex] using this identity:
[tex]\[
\sin(\theta + 10^\circ) = \cos(90^\circ - (\theta + 10^\circ))
\][/tex]
Simplifying the expression inside the cosine function, we get:
[tex]\[
\cos(90^\circ - (\theta + 10^\circ)) = \cos(90^\circ - \theta - 10^\circ) = \cos(80^\circ - \theta)
\][/tex]
Thus, our original equation becomes:
[tex]\[
\cos(80^\circ - \theta) = \cos(\theta)
\][/tex]
For [tex]\(\cos A = \cos B\)[/tex], it implies that:
[tex]\[
80^\circ - \theta = \theta \quad \text{or} \quad 80^\circ - \theta = 360^\circ n + 2\pi k - \theta \quad \text{(where \(n\) is an integer and \(k\) is zero or any integer)} \][/tex]
Solving the equation [tex]\(80^\circ - \theta = \theta\)[/tex]:
[tex]\[
80^\circ = 2\theta
\][/tex]
[tex]\[
\theta = 40^\circ
\][/tex]
Now, because we seek the value of [tex]\(\cos(\theta)\)[/tex],
[tex]\[
\cos(\theta) = \cos(40^\circ)
\][/tex]
For the given options:
- (b) [tex]\(80^\circ\)[/tex]
- (c) [tex]\(40^\circ\)[/tex]
- (d) [tex]\(100^\circ\)[/tex]
The correct answer that fits is indeed calculated as [tex]\(\theta = 40^\circ\)[/tex]. Therefore,
[tex]\(\cos(40^\circ)\)[/tex] is the value of [tex]\(\cos(\theta)\)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{40^\circ}
\][/tex]
