Use the given information to find the exact value of the exp

Use the given information to find the exact value of the expression sin alpha = 4/5, alpha lies in quadrant II, and cos beta = 2/5, beta lies in quadrant I. Find cos (alpha - beta) A. 8 + 3 Squareroot 21/25 B. -6 + 4 Squareroot 21/25 C. 6 - 4 Squareroot 21/25 D. 8 - 3 Squareroot 21/25

Solution

alpha = A

beta = B

Cos (A - B) = cos A*cos B + sin A*sin B

given that

sin A = 4/5 in II quadrant, than cos A = -sqrt (5^2 - 4^2)/5 = -3/5

cos B = 2/5 in I quadrant, sin B = +sqrt (5^2 - 2^2)/5 = (sqrt 21)/5

Using these values:

cos (A - B) = (-3/5)*(2/5) + (4/5)*((sqrt 21)/5)

= -6/25 + (4*sqrt 21)/25

= (-6 + 4*sqrt 21)/25

Correct option is B.