Find the Exact Value sin((13pi)/4)

Problem

sin((13*π)/4)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π until the angle is within the interval [0,2*π)

(13*π)/4−2*π=(13*π)/4−(8*π)/4

(13*π)/4−(8*π)/4=(5*π)/4

2. Identify the quadrant for the angle (5*π)/4 Since π<(5*π)/4<(3*π)/2 the angle lies in Quadrant III.

3. Determine the sign of the sine function in Quadrant III. In this quadrant, the ycoordinates are negative, so sin((5*π)/4) is negative.

4. Find the reference angle θ′ using the formula θ′=θ−π

θ′=(5*π)/4−π

θ′=π/4

5. Evaluate the sine of the reference angle.

sin(π/4)=√(,2)/2

6. Apply the sign from step 3 to the value found in step 5.

sin((5*π)/4)=−√(,2)/2

Final Answer

sin((13*π)/4)=−√(,2)/2

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