Question

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

Answer 1

To solve the equation \( \sin(2x) = \cos(4x + 12^\circ) \), we can make use of the co-function identity which states that \( \sin(\theta) = \cos(90^\circ - \theta) \).

This means we can set up the equation:

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

Simplifying this, we get:

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

Combine like terms:

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

\[
6x = 78^\circ
\]

Now, divide both sides by 6:

\[
x = 13^\circ
\]

Since we're looking for the acute angles in a right triangle, we now calculate \( 2x \):

\[
2x = 2 \times 13^\circ = 26^\circ
\]

Next, we can find \( 4x + 12^\circ \):

\[
4x + 12^\circ = 4 \times 13^\circ + 12^\circ = 52^\circ + 12^\circ = 64^\circ
\]

In a right triangle, the angles must sum to \( 180^\circ \) and one angle is \( 90^\circ \). Thus, the remaining two angles \( 26^\circ \) and \( 64^\circ \) are the acute angles.

The measure of the larger acute angle is:

\[
\boxed{64^\circ}
\]