Simplify sin(-315)

Problem

sin(−315^∘)

Solution

1. Apply the odd function property of the sine function, which states that sin(−θ)=−sin(θ)

sin(−315^∘)=−sin(315^∘)

2. Find the coterminal angle or reference angle for 315^∘ Since 315^∘ is in the fourth quadrant, the reference angle is 360^∘−315^∘=45^∘

315^∘=360^∘−45^∘

3. Determine the sign of the sine function in the fourth quadrant. Since sine is negative in the fourth quadrant, sin(315^∘)=−sin(45^∘)

−sin(315^∘)=−(−sin(45^∘))

4. Simplify the signs to show that the expression is equivalent to the sine of the reference angle.

−(−sin(45^∘))=sin(45^∘)

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

sin(45^∘)=√(,2)/2

Final Answer

sin(−315^∘)=√(,2)/2

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