Evaluate sin((3pi)/8)

Problem

sin((3*π)/8)

Solution

1. Identify the angle as half of a known angle from the unit circle. Since (3*π)/8=1/2⋅(3*π)/4 we can use the half-angle formula for sine.

2. Apply the formula for the half-angle of sine, which is sin(θ/2)=±√(,(1−cos(θ))/2)

3. Substitute θ=(3*π)/4 into the formula.

sin((3*π)/8)=√(,(1−cos((3*π)/4))/2)

4. Determine the sign based on the quadrant. Since (3*π)/8 is in the first quadrant (0<(3*π)/8<π/2, the sine value must be positive.

5. Evaluate the cosine value. We know that cos((3*π)/4)=−√(,2)/2

6. Simplify the expression inside the square root.

sin((3*π)/8)=√(,(1−(−√(,2)/2))/2)

sin((3*π)/8)=√(,(1+√(,2)/2)/2)

sin((3*π)/8)=√(,(2+√(,2))/2/2)

sin((3*π)/8)=√(,(2+√(,2))/4)

sin((3*π)/8)=√(,2+√(,2))/2

Final Answer

sin((3*π)/8)=√(,2+√(,2))/2

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