To solve for the angle [tex]\(\theta\)[/tex] in the given trigonometric equation [tex]\(\sin(\theta) = \cos(58^\circ)\)[/tex], let's proceed with the following steps:
### Step-by-Step Solution
1. Recognize Trigonometric Identity:
There is a fundamental trigonometric identity that connects sine and cosine:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
This identity tells us that the sine of an angle is equal to the cosine of its complement. Here, the angle [tex]\(\theta\)[/tex] and the angle [tex]\(58^\circ\)[/tex] are complements.
2. Set Up the Equation:
Given [tex]\(\sin(\theta) = \cos(58^\circ)\)[/tex], we can apply the identity:
[tex]\[
\sin(\theta) = \cos(58^\circ) = \cos(90^\circ - 32^\circ)
\][/tex]
This means:
[tex]\[
\theta = 90^\circ - \theta
\][/tex]
3. Determine the Complement:
Using the property of complementary angles, we recognize that we need to find the angle for which the sum when added to [tex]\(58^\circ\)[/tex] equals [tex]\(90^\circ\)[/tex]:
[tex]\[
\theta = 90^\circ - 58^\circ
\][/tex]
4. Calculate:
Subtract [tex]\(58^\circ\)[/tex] from [tex]\(90^\circ\)[/tex]:
[tex]\[
\theta = 32^\circ
\][/tex]
### Conclusion
The value of [tex]\(\theta\)[/tex] that satisfies the equation [tex]\(\sin(\theta) = \cos(58^\circ)\)[/tex] is:
[tex]\[
\theta = 32^\circ
\][/tex]
