Find the Exact Value tan((9pi)/8)

Problem

tan((9*π)/8)

Solution

1. Identify the angle in terms of a reference angle. Since (9*π)/8=π+π/8 the angle is in the third quadrant where the tangent function is positive.

2. Apply the periodicity of the tangent function, tan(π+θ)=tan(θ) to simplify the expression.

tan((9*π)/8)=tan(π/8)

3. Use the half-angle formula for tangent, tan(θ/2)=(1−cos(θ))/sin(θ) where θ=π/4

tan(π/8)=(1−cos(π/4))/sin(π/4)

4. Substitute the known values cos(π/4)=√(,2)/2 and sin(π/4)=√(,2)/2 into the formula.

tan(π/8)=(1−√(,2)/2)/√(,2)/2

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

tan(π/8)=(2−√(,2))/√(,2)

6. Rationalize the denominator by dividing each term in the numerator by √(,2)

tan(π/8)=2/√(,2)−√(,2)/√(,2)

tan(π/8)=√(,2)−1

Final Answer

tan((9*π)/8)=√(,2)−1

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