Evaluate cot((5pi)/4)

Problem

cot((5*π)/4)

Solution

1. Identify the angle (5*π)/4 in terms of its quadrant. Since π<(5*π)/4<(3*π)/2 the angle is in the third quadrant.

2. Determine the reference angle by subtracting π from the given angle.

(θ_ref)=(5*π)/4−π=π/4

3. Apply the definition of the cotangent function, which is cot(θ)=cos(θ)/sin(θ)

4. Evaluate the trigonometric values for the reference angle π/4

cos(π/4)=√(,2)/2

sin(π/4)=√(,2)/2

5. Determine the signs of the coordinates in the third quadrant. In the third quadrant, both cosine and sine are negative.

cos((5*π)/4)=−√(,2)/2

sin((5*π)/4)=−√(,2)/2

6. Calculate the final value by dividing the cosine value by the sine value.

cot((5*π)/4)=(−√(,2)/2)/(−√(,2)/2)=1

Final Answer

cot((5*π)/4)=1

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