Find the Exact Value sin(-(3pi)/8)

Problem

sin(−(3*π)/8)

Solution

1. Use the odd function property of the sine function, which states sin(−θ)=−sin(θ)

sin(−(3*π)/8)=−sin((3*π)/8)

2. Apply the half-angle formula for sine, sin(α/2)=√(,(1−cos(α))/2) where α=(3*π)/4

−sin((3*π)/8)=−√(,(1−cos((3*π)/4))/2)

3. Evaluate the cosine of (3*π)/4 which is in the second quadrant.

cos((3*π)/4)=−√(,2)/2

4. Substitute the value back into the half-angle expression.

−√(,(1−(−√(,2)/2))/2)=−√(,(1+√(,2)/2)/2)

5. Simplify the fraction inside the square root by multiplying the numerator and denominator by 2

−√(,(2+√(,2))/2/2)=−√(,(2+√(,2))/4)

6. Simplify the radical by taking the square root of the denominator.

−√(,2+√(,2))/2

Final Answer

sin(−(3*π)/8)=−√(,2+√(,2))/2

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