Find the Exact Value cos((8pi)/3)

Problem

cos((8*π)/3)

Solution

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

(8*π)/3−2*π=(8*π)/3−(6*π)/3

(8*π)/3−(6*π)/3=(2*π)/3

2. Identify the quadrant of the angle (2*π)/3 Since π/2<(2*π)/3<π the angle is in Quadrant II.

3. Determine the reference angle for (2*π)/3 in Quadrant II.

π−(2*π)/3=π/3

4. Apply the cosine sign for Quadrant II. In Quadrant II, the cosine function is negative.

cos((2*π)/3)=−cos(π/3)

5. Evaluate the exact value using the known value for cos(π/3)=1/2

−cos(π/3)=−1/2

Final Answer

cos((8*π)/3)=−1/2

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