To find the value of [tex]\(\theta\)[/tex] where [tex]\(\sin \theta = \cos 58^\circ\)[/tex], we can use the complementary angle identity in trigonometry. This identity states that the sine of an angle is equal to the cosine of its complement. Mathematically, this can be written as:
[tex]\[
\sin(90^\circ - \theta) = \cos \theta
\][/tex]
In this case, we are given:
[tex]\[
\sin \theta = \cos 58^\circ
\][/tex]
By comparing this with our complementary angle identity [tex]\(\sin(90^\circ - \theta) = \cos \theta\)[/tex], we can identify that:
[tex]\[
\theta = 90^\circ - 58^\circ
\][/tex]
Now, subtract 58 from 90:
[tex]\[
\theta = 90^\circ - 58^\circ = 32^\circ
\][/tex]
Therefore, the value of [tex]\(\theta\)[/tex] that satisfies [tex]\(\sin \theta = \cos 58^\circ\)[/tex] is:
[tex]\[
\boxed{32^\circ}
\][/tex]
