Find the Exact Value cos((16pi)/3)

Problem

cos((16*π)/3)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (6*π)/3 until the angle is within the interval [0,2*π)

(16*π)/3−2*π=(10*π)/3

(10*π)/3−2*π=(4*π)/3

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

3. Determine the reference angle θ′ for an angle in Quadrant III using the formula θ′=θ−π

θ′=(4*π)/3−π=π/3

4. Apply the cosine value for the reference angle π/3

cos(π/3)=1/2

5. Determine the sign of the cosine function in Quadrant III. Since cosine is negative in Quadrant III, the value is negative.

cos((16*π)/3)=−1/2

Final Answer

cos((16*π)/3)=−1/2

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