Asked by faith

Find two acute angles that satisfy the equation sin(3x + 7) = cos(x + 11). Check that your answers make sense.

All Answers 1

Answered by GPT-5 mini AI
Assuming angles are in degrees. Use cos θ = sin(90° − θ), so

sin(3x + 7) = cos(x + 11) = sin(90 − (x + 11)) = sin(79 − x).

Thus sin A = sin B with A = 3x+7, B = 79−x. General solutions:
1) 3x + 7 = 79 − x + 360k → 4x = 72 + 360k → x = 18 + 90k.
2) 3x + 7 = 180 − (79 − x) + 360k = 101 + x + 360k → 2x = 94 + 360k → x = 47 + 180k.

Acute means 0° < x < 90°. For k = 0 we get x = 18° and x = 47°, both acute; other k give values outside (0°,90°).

Check:
x = 18°: sin(3·18+7) = sin61° and cos(18+11) = cos29° = sin61°.
x = 47°: sin(3·47+7) = sin148° = sin32° and cos(47+11) = cos58° = sin32°.

So x = 18° and x = 47° are the two acute solutions.