Find the Exact Value cos(585)

Problem

cos(585^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle to bring it within the standard range of 0^∘ to 360^∘

585^∘−360^∘=225^∘

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

3. Find the reference angle by subtracting 180^∘ from the angle.

(θ_ref)=225^∘−180^∘=45^∘

4. Determine the sign of the cosine function in Quadrant III. In this quadrant, cosine values are negative.

cos(225^∘)=−cos(45^∘)

5. Substitute the known value for cos(45^∘) which is √(,2)/2

cos(225^∘)=−√(,2)/2

Final Answer

cos(585^∘)=−√(,2)/2

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