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(x/2+20)=cos(2x-25/2)

A) 25°
B) 33°
C) 36.5°
D) 53.5°

HINT: Sine and cosine of complementary angles are related

D) 53.5°

Step-by-step explanation:

Sine and cosine of complementary angles are equal:

sin θ = cos(90 − θ)

sin(x/2 + 20) = cos(2x − 25/2)

cos(90 − (x/2 + 20)) = cos(2x − 25/2)

90 − (x/2 + 20) = 2x − 25/2

90 − x/2 − 20 = 2x − 25/2

165/2 = 5x/2

5x = 165

x = 33

x/2 + 20 = 36.5

2x − 25/2 = 53.5

Since β > α, β = 53.5°.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240