Find the Exact Value 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 lies in Quadrant III.

2. Determine the sign of the cosine function in Quadrant III. In this quadrant, the xcoordinates are negative, so cos((4*π)/3) must be negative.

3. Calculate the reference angle (θ_r*e*ƒ) by subtracting π from the given angle.

(θ_r*e*ƒ)=(4*π)/3−π

(θ_r*e*ƒ)=π/3

4. Apply the reference angle to the cosine function, keeping the negative sign for Quadrant III.

cos((4*π)/3)=−cos(π/3)

5. Evaluate the exact value of cos(π/3) using the unit circle or a special right triangle.

cos(π/3)=1/2

Final Answer

cos((4*π)/3)=−1/2

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