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If cos(alpha) = 11/61 and sin(beta) = 4/5, where alpha and beta are positive acute angles, find sin((alpha - beta)/2) and cos((alpha + beta)/2).

1. sin((alpha-beta)/2) = 1/sqrt(122), cos((alpha+beta)/2) = 9/sqrt(122)
2. sin((alpha-beta)/2) = 9/sqrt(122), cos((alpha+beta)/2) = 1/sqrt(122)
3. sin((alpha-beta)/2) = 1/sqrt(61), cos((alpha+beta)/2) = 9/sqrt(61)
4. sin((alpha-beta)/2) = 3/5, cos((alpha+beta)/2) = 4/5

Correct answer: sin((alpha-beta)/2) = 1/sqrt(122), cos((alpha+beta)/2) = 9/sqrt(122)

Solution

With cos alpha = 11/61 (sin alpha = 60/61) and sin beta = 4/5 (cos beta = 3/5), the standard reduction yields sin((alpha-beta)/2) = 1/sqrt(122) and cos((alpha+beta)/2) = 9/sqrt(122).

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