Expand Using Sum/Difference Formulas sin(22.5)

Problem

sin(22.5)

Solution

1. Identify the angle as a difference of two common angles from the unit circle.

22.5=67.5−45

22.5=45−22.5

Alternatively, use the half-angle relationship:

22.5=45/2

2. Apply the formula for the sine half-angle identity, which is a specific case of sum/difference derivations.

sin(θ/2)=√(,(1−cos(θ))/2)

3. Substitute the value θ=45 into the formula.

sin(22.5)=√(,(1−cos(45))/2)

4. Evaluate the cosine of 45

cos(45)=√(,2)/2

5. Simplify the expression inside the square root.

sin(22.5)=√(,(1−√(,2)/2)/2)

sin(22.5)=√(,(2−√(,2))/2/2)

sin(22.5)=√(,(2−√(,2))/4)

sin(22.5)=√(,2−√(,2))/2

Final Answer

sin(22.5)=√(,2−√(,2))/2

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