Find the Exact Value cos((-4pi)/3)

Problem

cos(−(4*π)/3)

Solution

1. Apply the even-odd identity for the cosine function, which states cos(−θ)=cos(θ)

cos(−(4*π)/3)=cos((4*π)/3)

2. Find the reference angle by determining the distance to the nearest x-axis. Since (4*π)/3 is in the third quadrant, the reference angle is (4*π)/3−π

(θ_ref)=π/3

3. Determine the sign of the cosine function in the third quadrant. In the third quadrant, the x-coordinates are negative, so the cosine value is negative.

cos((4*π)/3)=−cos(π/3)

4. Evaluate the cosine of the reference angle using the unit circle or special right triangles.

cos(π/3)=1/2

5. Combine the sign and the value to find the final result.

cos(−(4*π)/3)=−1/2

Final Answer

cos(−(4*π)/3)=−1/2

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