Find the Exact Value cos((10pi)/3)

Problem

cos((10*π)/3)

Solution

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

(10*π)/3−2*π=(10*π)/3−(6*π)/3

(10*π)/3−(6*π)/3=(4*π)/3

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

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

θ′=(4*π)/3−π

θ′=π/3

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

cos((4*π)/3)=−cos(π/3)

5. Evaluate the cosine of the reference angle using known trigonometric values.

cos(π/3)=1/2

cos((4*π)/3)=−1/2

Final Answer

cos((10*π)/3)=−1/2

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