Find the Exact Value sin((5pi)/8)

Problem

sin((5*π)/8)

Solution

1. Identify the angle as a half-angle. Since (5*π)/8 is not a standard unit circle value, recognize that (5*π)/8=1/2⋅(5*π)/4

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

3. Determine the sign based on the quadrant. The angle (5*π)/8 is in the second quadrant (90^∘<112.5^∘<180^∘, where sine is positive.

4. Substitute θ=(5*π)/4 into the formula.

sin((5*π)/8)=√(,(1−cos((5*π)/4))/2)

5. Evaluate the cosine value. On the unit circle, cos((5*π)/4)=−√(,2)/2

sin((5*π)/8)=√(,(1−(−√(,2)/2))/2)

6. Simplify the expression inside the radical.

sin((5*π)/8)=√(,(1+√(,2)/2)/2)

sin((5*π)/8)=√(,(2+√(,2))/2/2)

sin((5*π)/8)=√(,(2+√(,2))/4)

sin((5*π)/8)=√(,2+√(,2))/2

Final Answer

sin((5*π)/8)=√(,2+√(,2))/2

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