Find the Exact Value cot(150)

Problem

cot(150^∘)

Solution

1. Identify the reference angle by finding the distance from the angle to the nearest x-axis. Since 150^∘ is in the second quadrant, the reference angle is 180^∘−150^∘=30^∘

2. Determine the sign of the cotangent function in the second quadrant. In the second quadrant, cosine is negative and sine is positive, so cot(θ)=cos(θ)/sin(θ) must be negative.

3. Apply the trigonometric identity for cotangent using the reference angle and the determined sign.

cot(150^∘)=−cot(30^∘)

4. Substitute the known value for cot(30^∘) Since cos(30^∘)=√(,3)/2 and sin(30^∘)=1/2 then cot(30^∘)=√(,3)

cot(150^∘)=−√(,3)

Final Answer

cot(150^∘)=−√(,3)

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