To solve the equation [tex]\(\sin (3x + 9)^\circ = \cos (5x - 7)^\circ\)[/tex], we can use the trigonometric identity that connects sine and cosine:
[tex]\[
\sin \theta = \cos (90^\circ - \theta)
\][/tex]
Applying this identity, we can rewrite [tex]\(\cos (5x - 7)^\circ\)[/tex] as:
[tex]\[
\cos (5x - 7)^\circ = \sin \left(90^\circ - (5x - 7)^\circ\right)
\][/tex]
So our original equation becomes:
[tex]\[
\sin (3x + 9)^\circ = \sin \left(90^\circ - (5x - 7)^\circ\right)
\][/tex]
Simplify the argument of the sine function on the right-hand side:
[tex]\[
90^\circ - (5x - 7)^\circ = 90^\circ - 5x + 7 = (97^\circ - 5x)
\][/tex]
Now the equation is:
[tex]\[
\sin (3x + 9)^\circ = \sin (97^\circ - 5x)
\][/tex]
Since the sine function is periodic and its values are equal when the angles differ by multiples of [tex]\(360^\circ\)[/tex] or are identical, we can equate the angles (since we are dealing with small angles in degrees, we can ignore the multiples of [tex]\(360^\circ\)[/tex]):
[tex]\[
3x + 9 = 97 - 5x
\][/tex]
To find [tex]\(x\)[/tex], we solve this equation step-by-step:
1. Add [tex]\(5x\)[/tex] to both sides:
[tex]\[
3x + 5x + 9 = 97
\][/tex]
2. Simplify the left-hand side:
[tex]\[
8x + 9 = 97
\][/tex]
3. Subtract 9 from both sides:
[tex]\[
8x = 97 - 9
\][/tex]
4. Simplify the right-hand side:
[tex]\[
8x = 88
\][/tex]
5. Finally, divide both sides by 8:
[tex]\[
x = \frac{88}{8}
\][/tex]
[tex]\[
x = 11
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{11}
\][/tex]
