To solve the problem of finding the value of [tex]\(\sin 34^\circ\)[/tex] given that [tex]\(\cos 56^\circ = 0.5592\)[/tex], we must utilize the trigonometric identity that relates sine and cosine. Specifically, we can use the co-function identity:
[tex]\[
\sin(90^\circ - \theta) = \cos(\theta)
\][/tex]
Let's apply this identity step-by-step:
1. Identify the relationship:
We need to find [tex]\(\sin 34^\circ\)[/tex]. From the co-function identity, we know that:
[tex]\[
\sin(90^\circ - \theta) = \cos(\theta)
\][/tex]
Set [tex]\(\theta = 56^\circ\)[/tex]. Then:
[tex]\[
\sin(90^\circ - 56^\circ) = \cos 56^\circ
\][/tex]
2. Simplify the expression:
Simplify [tex]\(90^\circ - 56^\circ\)[/tex]:
[tex]\[
90^\circ - 56^\circ = 34^\circ
\][/tex]
3. Substitute and solve:
Substitute the simplified value back into the identity:
[tex]\[
\sin 34^\circ = \cos 56^\circ
\][/tex]
4. Use the given value:
We've been given that [tex]\(\cos 56^\circ = 0.5592\)[/tex].
So,
[tex]\[
\sin 34^\circ = 0.5592
\][/tex]
Therefore, the value of [tex]\(\sin 34^\circ\)[/tex] is [tex]\(0.5592\)[/tex].
