Find the Exact Value sin(-pi/8)

Problem

sin(−π/8)

Solution

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

sin(−π/8)=−sin(π/8)

2. Identify the half-angle formula for sine, which is sin(θ/2)=±√(,(1−cos(θ))/2) Since π/8 is in the first quadrant, the value is positive.

sin(π/8)=√(,(1−cos(π/4))/2)

3. Substitute the known value for cos(π/4)=√(,2)/2 into the formula.

sin(π/8)=√(,(1−√(,2)/2)/2)

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

sin(π/8)=√(,(2−√(,2))/4)

5. Simplify the square root of the denominator.

sin(π/8)=√(,2−√(,2))/2

6. Combine the results by applying the negative sign from the first step.

−sin(π/8)=−√(,2−√(,2))/2

Final Answer

sin(−π/8)=−√(,2−√(,2))/2

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