Expand the Trigonometric Expression pi/8

Problem

π/8

Solution

1. Identify the expression as a common angle that can be represented as half of a standard angle, specifically 1/2⋅π/4

2. Apply the half-angle formula for trigonometric functions if the goal is to find the exact values of sin() cos() or tan() for this angle.

3. Calculate the cosine value using the identity cos(θ/2)=√(,(1+cos(θ))/2)

cos(π/8)=√(,(1+cos(π/4))/2)

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

cos(π/8)=√(,(1+√(,2)/2)/2)

5. Simplify the fraction inside the radical to find the exact expanded form.

cos(π/8)=√(,2+√(,2))/2

6. Calculate the sine value using the identity sin(θ/2)=√(,(1−cos(θ))/2)

sin(π/8)=√(,(1−√(,2)/2)/2)

7. Simplify the resulting expression.

sin(π/8)=√(,2−√(,2))/2

Final Answer

π/8=22.5

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