In a right triangle, the acute angles have the relationship

Question

In 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)

GPT-4o mini · Answer

To solve the equation $ \sin(x + 12°) = \cos(18° + 2x) $, we utilize the identity $ \cos(θ) = \sin(90° - θ) $. Rewriting the equation gives us: $$ \sin(x + 12°) = \sin(90° - (18° + 2x)) $$ This simplifies to: $$ \sin(x + 12°) = \sin(72° - 2x) $$ Setting the arguments equal to each other leads to two equations. The first gives us: $$ x + 12° = 72° - 2x $$ Solving this, we find: $$ 3x = 60° $$ $$ x = 20° $$ The second sine equation could lead to a different solution, but in this case, the values produced would not yield a valid angle when considering the constraints of a right triangle and the given equation. For the measure of the smaller angle in the triangle, we have $ x = 20° $ and the other acute angle would be $ 90° - 20° = 70° $. Thus, the measure of the smaller angle is $ 20° $. In summary, I set the equations from the given trigonometric identity, solved for $ x $, and determined the smaller angle from the right triangle's properties. The answers are $ x = 20° $ and the smaller angle is $ 20° $.

Answer

Can u make this copy and pasteable

GPT-4o mini · Answer

Sure! Here is the solution in a copy-pasteable format:

To solve the equation $ \sin(x + 12°) = \cos(18° + 2x) $, we utilize the identity $ \cos(θ) = \sin(90° - θ) $. Rewriting the equation gives us:

\[ \sin(x + 12°) = \sin(90° - (18° + 2x)) \]

This simplifies to:

\[ \sin(x + 12°) = \sin(72° - 2x) \]

Setting the arguments equal to each other leads to two equations. The first gives us:

\[ x + 12° = 72° - 2x \]

Solving this, we find:

\[ 3x = 60° \] \[ x = 20° \]

The second sine equation could lead to a different solution, but in this case, the values produced would not yield a valid angle when considering the constraints of a right triangle and the given equation.

For the measure of the smaller angle in the triangle, we have $ x = 20° $ and the other acute angle would be $ 90° - 20° = 70° $. Thus, the measure of the smaller angle is $ 20° $.

In summary, I set the equations from the given trigonometric identity, solved for $ x $, and determined the smaller angle from the right triangle's properties. The answers are $ x = 20° $ and the smaller angle is $ 20° $.

Feel free to copy and paste this wherever you need!