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°

14°
14°

28°
28°

16°

Answer 1

In a right triangle, the sine and cosine functions are related by the identity \(\sin(\theta) = \cos(90° - \theta)\). We can apply this identity to the equation given:

\[
\sin(3x + 10°) = \cos(x + 24°)
\]

Using the identity, we can equate the angles:

\[
3x + 10° = 90° - (x + 24°)
\]

Now simplify the right side:

\[
3x + 10° = 90° - x - 24°
\]
\[
3x + 10° = 66° - x
\]

Now, add \(x\) to both sides:

\[
3x + x + 10° = 66°
\]
\[
4x + 10° = 66°
\]

Next, subtract 10° from both sides:

\[
4x = 56°
\]

Now, divide both sides by 4:

\[
x = 14°
\]

So the value of \(x\) is:

\[
\boxed{14°}
\]