To solve the equation \( \sin(3x + 10^\circ) = \cos(2x - 10^\circ) \), we can use the complementary angle identity, which states that \( \sin(\theta) = \cos(90^\circ - \theta) \).
We can rewrite the equation as follows:
\[ \sin(3x + 10^\circ) = \cos(2x - 10^\circ) \]
Using the identity \( \sin(\theta) = \cos(90^\circ - \theta) \):
\[ \sin(3x + 10^\circ) = \cos(90^\circ - (3x + 10^\circ)) \]
This means that:
\[ \cos(90^\circ - (3x + 10^\circ)) = \cos(2x - 10^\circ) \]
Setting the angles equal to each other (considering \( \cos A = \cos B \) implies \( A = B \) or \( A = 360^\circ - B \)) gives us:
- First case: \[ 90^\circ - (3x + 10^\circ) = 2x - 10^\circ \]
Rearranging this gives us:
\[ 90^\circ - 3x - 10^\circ = 2x - 10^\circ \]
Combine like terms:
\[ 80^\circ = 5x \]
Solving for \(x\):
\[ x = \frac{80^\circ}{5} = 16^\circ \]
- Second case: \[ 90^\circ - (3x + 10^\circ) = 360^\circ - (2x - 10^\circ) \]
This could lead to extraneous solutions, so we'll be careful. Let's simplify:
\[ 90^\circ - 3x - 10^\circ = 360^\circ - 2x + 10^\circ \]
Which turns into:
\[ 80^\circ - 3x = 370^\circ - 2x \]
Rearranging gives:
\[ -3x + 2x = 370^\circ - 80^\circ \]
\[ -x = 290^\circ \]
\[ x = -290^\circ \] (not a valid solution in the context)
So the only valid solution is:
\[ x = 16^\circ \]
Thus, the correct answer from the given responses is 16°.