I Need Help, Please.

Question

Angles α and β are the two acute angles in a right triangle. Use the relationship between sine and cosine to find the value of β if β < α.

sin(2x + 2) = cos(3x + 8)

A) 16 B) 34 C) 56 D) 74

Answer 1

Sine and cosine are cofunctions which means that the sine of one acute angle is equal to the cosine of the other acute angle in the right triangle - aka sin(α) = cos (β).. or sin(β) = cos(α) depending on which angle is which - it doesn't matter for the purpose of solving.

Recall that the relationship between the acute angles in a right triangle is that they are complementary. Since all angles in a right triangle sum to 180°, the acute angles have to sum to 90°.

To solve for x, we take the measures of the two acute angles and sum them to 90°.

2x + 2 + 3x + 8 = 90°

5x + 10 = 90

5x = 80

x = 16

Now that we have x, we can plug in to find the measures of the angles. β has to be the measure of the smaller angle because it said β < α.

2(16) + 2 = 34°

3(16) + 8 = 56°.

Thus β = 34° choice B.

[If you want to check your answer, double check in your calculator that sin(34) = cos (56). Make sure you are in Degree mode.]