Question

In a right triangle, the acute angles have the relationship sin(x+12°)=cos(18°+2x) . What is the value of x? What is the measure of the smaller angle? Use 1–2 sentences to explain how you arrived at your answers.(2 points)

Answer 1

To solve the equation \(\sin(x + 12^\circ) = \cos(18^\circ + 2x)\), we can use the identity \(\sin(A) = \cos(90^\circ - A)\). Thus, we rewrite the equation as:

\[
\sin(x + 12^\circ) = \sin(90^\circ - (18^\circ + 2x)).
\]

This implies that:

\[
x + 12^\circ = 90^\circ - (18^\circ + 2x) \quad \text{or} \quad x + 12^\circ = 180^\circ - (90^\circ - (18^\circ + 2x)).
\]

Solving the first equation:

\[
x + 12^\circ = 90^\circ - 18^\circ - 2x,
\]
\[
x + 2x = 90^\circ - 18^\circ - 12^\circ,
\]
\[
3x = 60^\circ,
\]
\[
x = 20^\circ.
\]

The smaller angle in the triangle, considering the angles \(x\) and \(90^\circ - x\), is \(x = 20^\circ\). In conclusion, the value of \(x\) is \(20^\circ\), making the smaller angle \(20^\circ\). I arrived at this answer by using the trigonometric identity to simplify the sine and cosine relationship and then solving for \(x\).