Find the Exact Value cos((21pi)/4)

Problem

cos((21*π)/4)

Solution

1. Subtract multiples of 2*π to find the coterminal angle within the interval [0,2*π)

(21*π)/4=5*π+π/4

(21*π)/4−4*π=(21*π)/4−(16*π)/4

(21*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 is in Quadrant III.

3. Determine the sign of the cosine function in Quadrant III. In this quadrant, cos(θ) is negative.

4. Find the reference angle θ′

θ′=(5*π)/4−π

θ′=π/4

5. Evaluate the cosine of the reference angle and apply the negative sign.

cos(π/4)=√(,2)/2

cos((5*π)/4)=−√(,2)/2

Final Answer

cos((21*π)/4)=−√(,2)/2

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