Find the Exact Value cot(120 degrees )

Problem

cot(120^∘)

Solution

1. Identify the reference angle for 120^∘ by finding its distance from the 180^∘ mark on the x-axis.

(θ_ref)=180^∘−120^∘=60^∘

2. Determine the quadrant in which 120^∘ lies. Since 90^∘<120^∘<180^∘ the angle is in Quadrant II.

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

cot(120^∘)=−cot(60^∘)

4. Apply the trigonometric identity cot(θ)=cos(θ)/sin(θ) using the known values for 60^∘

cos(60^∘)=1/2

sin(60^∘)=√(,3)/2

5. Calculate the value by dividing the cosine by the sine.

cot(60^∘)=(1/2)/(√(,3)/2)=1/√(,3)

6. Rationalize the denominator and apply the negative sign from step 3.

cot(120^∘)=−1/√(,3)⋅√(,3)/√(,3)=−√(,3)/3

Final Answer

cot(120^∘)=−√(,3)/3

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