Find the Value Using the Unit Circle sin(22.5)

Problem

sin(22.5)

Solution

1. Identify the angle as a half-angle of a known value on the unit circle, 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, using the unit circle value cos(45)=√(,2)/2

sin(22.5)=√(,(1−√(,2)/2)/2)

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

sin(22.5)=√(,(2−√(,2))/2/2)

6. Divide by 2 to reach the final radical form.

sin(22.5)=√(,(2−√(,2))/4)

7. 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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