Find the Value Using the Unit Circle sin(pi/8)

Problem

sin(π/8)

Solution

1. Identify the angle π/8 as half of a known angle on the unit circle, specifically π/4

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

3. Substitute θ=π/4 into the formula.

4. Determine the sign based on the quadrant. Since π/8 is in the first quadrant, the sine value must be positive.

5. Evaluate the cosine of π/4 which is √(,2)/2

sin(π/8)=√(,(1−√(,2)/2)/2)

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

sin(π/8)=√(,(2−√(,2))/2/2)

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

sin(π/8)=√(,(2−√(,2))/4)

8. Simplify the square root of the denominator.

sin(π/8)=√(,2−√(,2))/2

Final Answer

sin(π/8)=√(,2−√(,2))/2

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