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 equation \( \sin(2x) = \cos(4x + 12^\circ) \), we can use the identity that relates sine and cosine:

\[
\sin(θ) = \cos(90^\circ - θ)
\]

This means we can rewrite the equation as:

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

Now, solve for \( x \):

\[
2x = 90^\circ - 4x - 12^\circ
\]
\[
2x + 4x = 90^\circ - 12^\circ
\]
\[
6x = 78^\circ
\]
\[
x = \frac{78^\circ}{6} = 13^\circ
\]

Now that we have \( x \), we can find the measures of the acute angles in the right triangle. The angles are:

- One angle is \( x = 13^\circ \).
- The second angle, which we can denote as \( y \), can be found using the fact that the sum of angles in a triangle is \( 180^\circ \), and one angle is \( 90^\circ \):

\[
y = 90^\circ - x = 90^\circ - 13^\circ = 77^\circ
\]

Thus, the larger angle in the right triangle is \( 77^\circ \).

So the measure of the larger angle is:

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