Find the Exact Value sin(585)

Problem

sin(585^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle to bring it within the 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. Calculate 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 sine function in Quadrant III. In this quadrant, sine is negative.

sin(225^∘)=−sin(45^∘)

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

−sin(45^∘)=−√(,2)/2

Final Answer

sin(585^∘)=−√(,2)/2

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