Find the Exact Value cot(240)

Problem

cot(240^∘)

Solution

1. Identify the reference angle by determining the quadrant in which 240^∘ lies. Since 180^∘<240^∘<270^∘ the angle is in Quadrant III.

2. Calculate the reference angle θ^′ using the formula for Quadrant III: θ^′=θ−180^∘

θ^′=240^∘−180^∘=60^∘

3. Determine the sign of the cotangent function in Quadrant III. In Quadrant III, both sine and cosine are negative, which means cot(θ)=cos(θ)/sin(θ) is positive.

4. Evaluate the cotangent of the reference angle using known trigonometric values.

cot(60^∘)=cos(60^∘)/sin(60^∘)

cot(60^∘)=(1/2)/(√(,3)/2)

cot(60^∘)=1/√(,3)

5. Rationalize the denominator to express the value in standard form.

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot(240^∘)=√(,3)/3

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