Find the Exact Value tan(pi/8)

Problem

tan(π/8)

Solution

1. Identify the half-angle identity for tangent that will simplify the calculation. We use the formula:

tan(θ/2)=(1−cos(θ))/sin(θ)

2. Substitute θ=π/4 into the formula, since π/8 is half of π/4

tan(π/8)=(1−cos(π/4))/sin(π/4)

3. Evaluate the trigonometric values for π/4

cos(π/4)=√(,2)/2

sin(π/4)=√(,2)/2

4. Plug these values back into the expression.

tan(π/8)=(1−√(,2)/2)/√(,2)/2

5. Simplify the complex fraction by multiplying the numerator and denominator by 2

tan(π/8)=(2−√(,2))/√(,2)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

tan(π/8)=((2−√(,2))√(,2))/(√(,2)⋅√(,2))

tan(π/8)=(2√(,2)−2)/2

7. Divide each term in the numerator by 2

tan(π/8)=√(,2)−1

Final Answer

tan(π/8)=√(,2)−1

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