To solve the equation [tex]\(\sin(3x + 13^\circ) = \cos(4x)\)[/tex], we need to find an [tex]\(x\)[/tex] that satisfies this trigonometric identity.
First, recall the co-function identity: [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex]. Using this, we can rewrite [tex]\(\cos(4x)\)[/tex] in terms of sine:
[tex]\[
\cos(4x) = \sin\left(90^\circ - 4x\right)
\][/tex]
Thus, the given equation transforms to:
[tex]\[
\sin(3x + 13^\circ) = \sin(90^\circ - 4x)
\][/tex]
For this equality to hold, the arguments of the sine functions must either be equal or differ by multiples of [tex]\(180^\circ\)[/tex], due to the periodicity and symmetry of the sine function. Thus, we have two cases to consider:
Case 1:
[tex]\[
3x + 13^\circ = 90^\circ - 4x + 360^\circ k \quad \text{for any integer } k
\][/tex]
Simplifying,
[tex]\[
3x + 4x = 90^\circ - 13^\circ + 360^\circ k
\][/tex]
[tex]\[
7x = 77^\circ + 360^\circ k
\][/tex]
[tex]\[
x = 11^\circ + 51.43^\circ k
\][/tex]
By considering integer values for [tex]\(k\)[/tex], we find that [tex]\(x = 11^\circ\)[/tex] is a possible solution.
Case 2:
[tex]\[
3x + 13^\circ = 180^\circ - (90^\circ - 4x) + 360^\circ k
\][/tex]
[tex]\[
3x + 13^\circ = 90^\circ + 4x + 360^\circ k
\][/tex]
Simplifying,
[tex]\[
3x - 4x = 90^\circ - 13^\circ + 360^\circ k
\][/tex]
[tex]\[
-x = 77^\circ + 360^\circ k
\][/tex]
[tex]\[
x = -77^\circ - 360^\circ k
\][/tex]
This scenario does not yield an [tex]\(x\)[/tex] within the provided answer choices as valid solutions.
Given the solutions derived from comprehensive analysis, [tex]\(x = 11^\circ\)[/tex] matches with the given answer choices.
Thus, the correct answer is:
[tex]\[
\boxed{11^\circ}
\][/tex]
