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