Find the Exact Value sin(67.5)

Problem

sin(67.5)

Solution

1. Identify the angle as a half-angle of a known value from the unit circle.

67.5=135/2

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

sin(67.5)=√(,(1−cos(135))/2)

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

sin(67.5)>0

4. Substitute the known value cos(135)=−√(,2)/2 into the formula.

sin(67.5)=√(,(1−(−√(,2)/2))/2)

5. Simplify the expression inside the square root by finding a common denominator in the numerator.

sin(67.5)=√(,(2+√(,2))/2/2)

6. Divide the fraction by 2.

sin(67.5)=√(,(2+√(,2))/4)

7. Simplify the radical by taking the square root of the denominator.

sin(67.5)=√(,2+√(,2))/2

Final Answer

sin(67.5)=√(,2+√(,2))/2

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