Find the Exact Value cot(135)

Problem

cot(135)

Solution

1. Identify the angle in the coordinate plane. The angle 135 is located in the second quadrant.

2. Determine the reference angle. For an angle θ in the second quadrant, the reference angle θ′ is calculated as 180−θ

θ′=180−135=45

3. Apply the definition of the cotangent function. The cotangent of an angle is the ratio of the cosine to the sine.

cot(135)=cos(135)/sin(135)

4. Determine the signs of the trigonometric functions in the second quadrant. In the second quadrant, cosine is negative and sine is positive.

cos(135)=−cos(45)

sin(135)=sin(45)

5. Substitute the known values for 45 We know that cos(45)=√(,2)/2 and sin(45)=√(,2)/2

cot(135)=(−√(,2)/2)/√(,2)/2

6. Simplify the expression to find the final value.

cot(135)=−1

Final Answer

cot(135)=−1

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