Let α and β be first quadrant angles with cos ( α ) = √11/ 6 and sin ( β ) = √10/ 9 . Find sin ( α − β ) .

Answer 1

First, sin(a - b) = sin a cos b - cos a sin b. Also we know that sin²x + cos ²x = 1, so:

sin²x = 1- cos²x, which means sin x = √(1- cos²x)

cos²x = 1 - sin ²x, which means cos x = √(1-sin²x)

With those housekeeping measures out of the way:

We know cos a and sin b, but in order to get sin(a-b) we also need sin a and cos b:

sin a = √(1-cos²(a) = √1 - (√11/6)² = √1 - 11/36 = √25/36 = 5/6

cos b = √(1-sin²(b)) = √(1 - (√10 / 9)²)= √1 - 10 / 81 = √71/81 = √71/9

Therefore sin(a-b) = sin a cos b - cos a sin b

= (5/6)(√71/9) - (√11/6)(√10/9)

= (5√71/54) - (√110 /54)

= (5√71 - √110)/54