Find the Exact Value sin(22.5 degrees )

Problem

sin(22.5)

Solution

1. Identify the angle as half of a known special angle, since 22.5=45/2

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 22.5 is in the first quadrant, the sine value must be positive.

4. Substitute θ=45 into the formula:

sin(22.5)=√(,(1−cos(45))/2)

5. Substitute the known value cos(45)=√(,2)/2

sin(22.5)=√(,(1−√(,2)/2)/2)

6. Simplify the numerator by finding a common denominator:

sin(22.5)=√(,(2−√(,2))/2/2)

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

sin(22.5)=√(,(2−√(,2))/4)

8. Simplify the square root of the denominator:

sin(22.5)=√(,2−√(,2))/2

Final Answer

sin(22.5)=√(,2−√(,2))/2

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