Find the Exact Value cos((23pi)/4)

Problem

cos((23*π)/4)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (8*π)/4 from the given angle to bring it within the interval [0,2*π)

(23*π)/4−2*π=(15*π)/4

(15*π)/4−2*π=(7*π)/4

2. Determine the quadrant of the angle (7*π)/4 Since (3*π)/2<(7*π)/4<2*π the angle lies in Quadrant IV.

3. Identify the reference angle for (7*π)/4 in Quadrant IV.

(θ_ref)=2*π−(7*π)/4

(θ_ref)=π/4

4. Determine the sign of the cosine function in Quadrant IV. In the fourth quadrant, the cosine of an angle is positive.

cos((23*π)/4)=cos(π/4)

5. Evaluate the exact value using the unit circle or special right triangles.

cos(π/4)=√(,2)/2

Final Answer

cos((23*π)/4)=√(,2)/2

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