Question

For the acute angles in a right triangle, sin(3x+10°)=cos(x+24°)
. What is the value of x?(1 point)
Responses

31°
31°

28°
28°

16°
16°

14°

Answer 1

To solve the equation \( \sin(3x + 10^\circ) = \cos(x + 24^\circ) \), we can use the identity that states \( \sin(\theta) = \cos(90^\circ - \theta) \).

This means we can rewrite the equation as follows:

\[
\sin(3x + 10^\circ) = \cos(x + 24^\circ) \implies 3x + 10^\circ = 90^\circ - (x + 24^\circ)
\]

Now, simplify the right side:

\[
3x + 10^\circ = 90^\circ - x - 24^\circ
\]

Combine the constants on the right side:

\[
3x + 10^\circ = 66^\circ - x
\]

Next, we will isolate \(x\):

\[
3x + x = 66^\circ - 10^\circ
\]
\[
4x = 56^\circ
\]
\[
x = \frac{56^\circ}{4} = 14^\circ
\]

Thus, the value of \(x\) is:

\[
\boxed{14^\circ}
\]