Expand Using Sum/Difference Formulas sin(pi/8)

Problem

sin(π/8)

Solution

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

π/8=1/2⋅π/4

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

sin(π/8)=√(,(1−cos(π/4))/2)

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

sin(π/8)=√(,(1−√(,2)/2)/2)

4. Simplify the numerator by finding a common denominator.

sin(π/8)=√(,(2−√(,2))/2/2)

5. Divide the fraction inside the square root.

sin(π/8)=√(,(2−√(,2))/4)

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

sin(π/8)=√(,2−√(,2))/2

Final Answer

sin(π/8)=√(,2−√(,2))/2

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