Simplify sin(112.5)

Problem

sin(112.5)

Solution

1. Identify the angle as a half-angle of a known special angle.

θ=112.5

2*θ=225

2. Apply the half-angle formula for sine, which is sin(α/2)=±√(,(1−cos(α))/2)

sin(112.5)=±√(,(1−cos(225))/2)

3. Determine the sign based on the quadrant. Since 112.5 is in Quadrant II, the sine value must be positive.

sin(112.5)=√(,(1−cos(225))/2)

4. Evaluate the cosine of 225 Since 225 is in Quadrant III with a reference angle of 45 the cosine is negative.

cos(225)=−√(,2)/2

5. Substitute the value back into the formula.

sin(112.5)=√(,(1−(−√(,2)/2))/2)

6. Simplify the expression inside the radical.

sin(112.5)=√(,(1+√(,2)/2)/2)

sin(112.5)=√(,(2+√(,2))/2/2)

sin(112.5)=√(,(2+√(,2))/4)

sin(112.5)=√(,2+√(,2))/2

Final Answer

sin(112.5)=√(,2+√(,2))/2

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