alleryStart learning →Evaluate sin(45 degrees - 30 degrees) exactly using the difference formula for sine.
A(sqrt(2) - sqrt(6)) / 4
B(sqrt(6) - sqrt(2)) / 4
C(sqrt(6) + sqrt(2)) / 4
D(sqrt(3) - sqrt(2)) / 4
Answer & Solution
Correct answer: B. (sqrt(6) - sqrt(2)) / 4
1. Write sin(a - b) = sin a cos b - cos a sin b with a = 45 degrees and b = 30 degrees.
2. The needed values are sin 45 = sqrt(2)/2, cos 30 = sqrt(3)/2, cos 45 = sqrt(2)/2 and sin 30 = 1/2.
3. The first product is (sqrt(2)/2)(sqrt(3)/2) = sqrt(6)/4.
4. The second product is (sqrt(2)/2)(1/2) = sqrt(2)/4.
5. Subtracting gives (sqrt(6) - sqrt(2)) / 4.
6. Reversing the order of subtraction would give the negative of this, which cannot be right since 15 degrees has a positive sine.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.2 Sum and Difference Identities_
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