Find the Exact Value cot(120)

Problem

cot(120^∘)

Solution

1. Identify the reference angle by finding the distance from the angle to the x-axis. Since 120^∘ is in the second quadrant, the reference angle is 180^∘−120^∘=60^∘

2. Determine the sign of the cotangent function in the second quadrant. In the second quadrant, x is negative and y is positive, so cot(θ)=x/y is negative.

3. Apply the trigonometric identity for cotangent using the reference angle.

cot(120^∘)=−cot(60^∘)

4. Substitute the known exact value for cot(60^∘) which is 1/√(,3) or √(,3)/3

cot(60^∘)=cos(60^∘)/sin(60^∘)=(1/2)/(√(,3)/2)=1/√(,3)

5. Rationalize the denominator to express the final value.

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot(120^∘)=−√(,3)/3

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