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

Answer 1

To solve the equation \( \sin(x + 12^\circ) = \cos(18^\circ + 2x) \), we can use the identity \(\sin(A) = \cos(90^\circ - A)\). This leads us to set up the equation:

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

Simplifying, we get:

\[ x + 12^\circ = 90^\circ - 18^\circ - 2x \ x + 12^\circ = 72^\circ - 2x \ 3x = 60^\circ \ x = 20^\circ \]

The smaller angle in the triangle, which is \(x\), measures \(20^\circ\). I arrived at this value by using the sine-cosine identity and solving the resulting linear equation for \(x\).