Question

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.

Answer 1

To solve the equation \( \sin(x + 12°) = \cos(18° + 2x) \), we can use the identity \( \cos(A) = \sin(90° - A) \). Hence, we rewrite the equation as \( \sin(x + 12°) = \sin(72° - 2x) \).

Setting the angles equal to each other gives us two equations to solve:
1. \( x + 12° = 72° - 2x \)
2. \( x + 12° = 180° - (72° - 2x) \)

Solving the first equation:
1. \( x + 2x = 72° - 12° \)
2. \( 3x = 60° \)
3. \( x = 20° \)

Now, for the second equation:
1. Substituting and simplifying provides no further valid acute angles since it leads to an invalid scenario given the sine function's properties.

Therefore, we confirm \( x = 20° \) with the smaller angle of the triangle being \( 20° \) and the other acute angle (which can be found using \( 90° - 20° \)) being \( 70° \).

In conclusion, \( x = 20° \) and the smaller angle in the triangle is also \( 20° \) because that is one of the two acute angles obtained from the original condition.

Answer 2

Please use 1-2 sentences to explain how you arrived at these answers.