Question

For the acute angles in a right triangle, sin(2x)=cos(4x+12°) . What is the measure of the larger angle?(1 point)

Answer 1

To solve the problem where \( \sin(2x) = \cos(4x + 12^\circ) \), we can use the complementary angle identity, which states that \( \sin(A) = \cos(90^\circ - A) \).

From this identity, we can rewrite the equation as:

\[
\sin(2x) = \cos(4x + 12^\circ) \implies 2x = 90^\circ - (4x + 12^\circ)
\]

Now, let's simplify this equation:

\[
2x = 90^\circ - 4x - 12^\circ
\]

Combine the terms on the right side:

\[
2x = 78^\circ - 4x
\]

Now, add \( 4x \) to both sides:

\[
2x + 4x = 78^\circ
\]

This simplifies to:

\[
6x = 78^\circ
\]

Now, divide both sides by 6:

\[
x = \frac{78^\circ}{6} = 13^\circ
\]

Since \( x \) represents one of the acute angles in the right triangle, the angles of the triangle are:

- One angle: \( x = 13^\circ \)
- The larger acute angle: \( 90^\circ - x = 90^\circ - 13^\circ = 77^\circ \)

Therefore, the measure of the larger angle is:

\[
\boxed{77^\circ}
\]