Find the Exact Value sin((9pi)/8)

Problem

sin((9*π)/8)

Solution

1. Identify the angle and its quadrant. The angle (9*π)/8 is in the third quadrant because π<(9*π)/8<(3*π)/2 In the third quadrant, the sine function is negative.

2. Determine the reference angle. The reference angle is (9*π)/8−π=π/8

3. Apply the half-angle formula for sine. The formula is sin(θ/2)=±√(,(1−cos(θ))/2) Let θ/2=(9*π)/8 which means θ=(9*π)/4

4. Evaluate the cosine of the doubled angle. Since (9*π)/4 is coterminal with π/4 we have cos((9*π)/4)=cos(π/4)=√(,2)/2

5. Substitute the value into the half-angle formula. Since the sine of an angle in the third quadrant is negative, we choose the negative root.

sin((9*π)/8)=−√(,(1−√(,2)/2)/2)

6. Simplify the expression inside the radical. Multiply the numerator and denominator by 2

sin((9*π)/8)=−√(,(2−√(,2))/4)

7. Extract the square root from the denominator.

sin((9*π)/8)=−√(,2−√(,2))/2

Final Answer

sin((9*π)/8)=−√(,2−√(,2))/2

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