Simplify sin(67.5)

Problem

sin(67.5)

Solution

1. Identify the angle as a half-angle of a known special angle, noting that 67.5=135/2

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

3. Determine the sign based on the quadrant. Since 67.5 is in the first quadrant, the sine value is positive.

4. Substitute θ=135 into the formula.

sin(67.5)=√(,(1−cos(135))/2)

5. Evaluate the cosine of 135 which is −√(,2)/2

sin(67.5)=√(,(1−(−√(,2)/2))/2)

6. Simplify the expression inside the radical by finding a common denominator in the numerator.

sin(67.5)=√(,(2+√(,2))/2/2)

7. Divide by 2 to reach the final simplified form.

sin(67.5)=√(,(2+√(,2))/4)

sin(67.5)=√(,2+√(,2))/2

Final Answer

sin(67.5)=√(,2+√(,2))/2

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