Find the Exact Value cot(585)

Problem

cot(585^∘)

Solution

1. Find the coterminal angle by subtracting multiples of 360^∘ until the angle is between 0^∘ and 360^∘

585^∘−360^∘=225^∘

2. Identify the quadrant for the angle 225^∘ Since 180^∘<225^∘<270^∘ the angle lies in Quadrant III.

3. Determine the reference angle for an angle in Quadrant III using the formula (θ_ref)=θ−180^∘

225^∘−180^∘=45^∘

4. Determine the sign of the cotangent function in Quadrant III. In Quadrant III, cotangent is positive.

cot(225^∘)=cot(45^∘)

5. Evaluate the exact value using the known trigonometric value for 45^∘

cot(45^∘)=1

Final Answer

cot(585^∘)=1

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