Find the Exact Value cos((2pi)/3)

Problem

cos((2*π)/3)

Solution

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

2. Determine the sign of the cosine function in the second quadrant. In the second quadrant, the xcoordinate is negative, so cos(θ) is negative.

3. Calculate the reference angle θ′ The reference angle for an angle in the second quadrant is π−θ

θ′=π−(2*π)/3

θ′=π/3

4. Evaluate the cosine of the reference angle. Using the unit circle or special triangles, the value of cos(π/3) is 1/2

5. Apply the sign from step 2 to the value from step 4.

cos((2*π)/3)=−cos(π/3)

cos((2*π)/3)=−1/2

Final Answer

cos((2*π)/3)=−1/2

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