n 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

In the given right triangle, using the co-function identity \( \sin(\theta) = \cos(90^\circ - \theta) \), we can rewrite \( \sin(x + 12^\circ) = \cos(18^\circ + 2x) \) as \( \sin(x + 12^\circ) = \sin(72^\circ - 2x) \). Setting the angles equal gives us \( x + 12^\circ = 72^\circ - 2x \), leading to \( 3x = 60^\circ \), resulting in \( x = 20^\circ \). Therefore, the smaller angle in the triangle is \( 20^\circ \) since it must be less than \( 90^\circ \) and we can verify \( 12^\circ \) (the other acute angle) is less.