Find the Exact Value sin(-75)

Problem

sin(−75)

Solution

1. Apply the odd function identity for sine, which states that sin(−θ)=−sin(θ)

sin(−75)=−sin(75)

2. Rewrite the angle as a sum of two special angles from the unit circle, such as 45 and 30

−sin(75)=−sin(45+30)

3. Apply the sum identity for sine, sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

−sin(45+30)=−(sin(45)*cos(30)+cos(45)*sin(30))

4. Substitute the exact values for the sine and cosine of the special angles.

−(√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2)

5. Simplify the expression by multiplying the fractions and combining the terms over a common denominator.

−(√(,6)/4+√(,2)/4)

−(√(,6)+√(,2))/4

Final Answer

sin(−75)=−(√(,6)+√(,2))/4

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