In a right triangle, the acute angles have the relationship sin(x+12°)=cos(18°+2x)

Question

sin
⁡
(
𝑥
+
12
°
)
=
cos
⁡
(
18
°
+
2
𝑥
)
. 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.

Answer 1

In a right triangle, we have the relationship \( \sin(x + 12^\circ) = \cos(18^\circ + 2x) \). Using the identity \( \cos(A) = \sin(90^\circ - A) \), we can rewrite the equation as \( \sin(x + 12^\circ) = \sin(72^\circ - 2x) \). Setting the angles equal gives us two equations:

1. \( x + 12^\circ = 72^\circ - 2x \)
2. \( x + 12^\circ = 180^\circ - (72^\circ - 2x) \) (not applicable here since both angles are acute in a right triangle).

Solving \( x + 12^\circ = 72^\circ - 2x \):

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

Thus, the smaller angle, \( x \), is \( 20^\circ \).

To verify, calculate \( x + 12^\circ = 32^\circ \) and \( 18^\circ + 2x = 58^\circ \); since \( \sin(32^\circ) = \cos(58^\circ) \) holds true, the values are confirmed.

The smaller angle is \( 20^\circ \).