Simplify cos((4pi)/3)

Problem

cos((4*π)/3)

Solution

1. Identify the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle is in the third quadrant.

2. Determine the reference angle. The reference angle θ′ for an angle in the third quadrant is calculated as θ′=θ−π

θ′=(4*π)/3−π=π/3

3. Apply the sign for the cosine function in the third quadrant. In the third quadrant, the cosine of an angle is negative.

cos((4*π)/3)=−cos(π/3)

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

cos((4*π)/3)=−1/2

Final Answer

cos((4*π)/3)=−1/2

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