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.