im confused here like i know what formula to use but im not getting the right answer

Question

Let  and  be first quadrant angles with  and .Find .

Answer 1

Use

To do so requires (obviously) having both the sine and cosine of both α and β.

For angle α:

We are told that the cos(α) = √11/4 and we know that cos(α) = adjacent/hypotenuse therefore we can say the adjacent side is √11 and the hypotenuse is 4. Using the Pythagorean Theorem, (√11)² + (opposite)² = 4² so opposite = √(16 - 11) = √5 which means sin(α) = √5/4

For angle β:

We are told that the sin(β) = √10/8 and we know that sin(β) = opposite/hypotenuse therefore we can say the opposite side is √10 and the hypotenuse is 8. Using the Pythagorean Theorem, (√10)² + (adjacent)² = 8² so adjacent = √(64 - 10) = √54 = 3√6 which means cos(β) = 3√6/8

So since sin(α - β) = sin(α)cos(β) - cos(α)sin(β) then:

sin(α - β) = (√5/4)(3√6/8) - (√11/4)(√10/8)

sin(α - β) = 3√30/32 - √110/32 = (3√30 - √110)/32