Suppose that sin alpha = 6/10 and sin beta = 1/2, where pi/2<beta<alpha<pi. Find sin(alpha-beta)

Question

Simplify the expression

Answer 1

sin A = 6/10 = 3/5 --> opposite =3 and hypotenuse = 5

Pythagorean says adjacent is sqrt(5^2 - 3^2) = sqrt(25-9) = sqrt(16)=4

so then cos A = 4/5

sin B = 1/2 --> opposite = 1 and hypotenuse = 2

Pythagorean says adjacent is sqrt(2^2 - 1^2) = sqrt(4-1) = sqrt(3)

so cos B = sqrt(3)/2

Angle subtraction formula for

sin(A-b) = sinA cos B - sin B cos A

= (3/5)(sqrt(3))/2 - (1/2) (4/5)

= 3*sqrt(3)/10 - 4/10

= (3*sqrt(3)-4)/10