Find the Exact Value tan(-(3pi)/8)

Problem

tan(−(3*π)/8)

Solution

1. Use the odd function property of the tangent function, which states tan(−θ)=−tan(θ)

tan(−(3*π)/8)=−tan((3*π)/8)

2. Identify the half-angle by recognizing that (3*π)/8 is half of (3*π)/4

(3*π)/8=1/2⋅(3*π)/4

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

−tan((3*π)/8)=−(1−cos((3*π)/4))/sin((3*π)/4)

4. Substitute the known values for the sine and cosine of (3*π)/4 which are sin((3*π)/4)=√(,2)/2 and cos((3*π)/4)=−√(,2)/2

−(1−(−√(,2)/2))/√(,2)/2=−(1+√(,2)/2)/√(,2)/2

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

−(2+√(,2))/√(,2)

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

−(2/√(,2)+√(,2)/√(,2))=−(√(,2)+1)

7. Distribute the negative sign to find the final exact value.

−1−√(,2)

Final Answer

tan(−(3*π)/8)=−1−√(,2)

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