Find the Exact Value sin(1305)

Problem

sin(1305^∘)

Solution

1. Find the coterminal angle by dividing 1305^∘ by 360^∘ to determine how many full rotations are in the angle.

1305÷360=3.625

2. Subtract the full rotations from the original angle to find an angle between 0^∘ and 360^∘

1305^∘−3*(360^∘)=1305^∘−1080^∘

1305^∘−1080^∘=225^∘

3. Identify the quadrant for 225^∘ Since 180^∘<225^∘<270^∘ the angle lies in Quadrant III, where the sine function is negative.

sin(1305^∘)=sin(225^∘)

4. Determine the reference angle by subtracting 180^∘ from the angle.

(θ_ref)=225^∘−180^∘

(θ_ref)=45^∘

5. Apply the sine value for the reference angle and attach the appropriate sign for Quadrant III.

sin(225^∘)=−sin(45^∘)

sin(45^∘)=√(,2)/2

Final Answer

sin(1305^∘)=−√(,2)/2

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