Find the Exact Value 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 (θ_r) by subtracting π from the given angle.

(θ_r)=(5*π)/4−π

(θ_r)=π/4

3. Apply the definition of the cotangent function, which is the ratio of the cosine to the sine.

cot((5*π)/4)=cos((5*π)/4)/sin((5*π)/4)

4. Evaluate the trigonometric values in the third quadrant, where both sine and cosine are negative.

cos((5*π)/4)=−√(,2)/2

sin((5*π)/4)=−√(,2)/2

5. Simplify the ratio to find the final value.

cot((5*π)/4)=(−√(,2)/2)/(−√(,2)/2)

cot((5*π)/4)=1

Final Answer

cot((5*π)/4)=1

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