Find the Exact Value tan((7pi)/8)

Problem

tan((7*π)/8)

Solution

1. Identify the angle as a half-angle. Since (7*π)/8=1/2⋅(7*π)/4 we can use the tangent half-angle formula.

2. Select the half-angle formula for tangent:

tan(θ/2)=(1−cos(θ))/sin(θ)

3. Substitute θ=(7*π)/4 into the formula:

tan((7*π)/8)=(1−cos((7*π)/4))/sin((7*π)/4)

4. Determine the values of the trigonometric functions at (7*π)/4 This angle is in the fourth quadrant where cosine is positive and sine is negative:

cos((7*π)/4)=√(,2)/2

sin((7*π)/4)=−√(,2)/2

5. Plug these values back into the expression:

tan((7*π)/8)=(1−√(,2)/2)/(−√(,2)/2)

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

tan((7*π)/8)=(2−√(,2))/(−√(,2))

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

tan((7*π)/8)=((2−√(,2))√(,2))/(−√(,2)√(,2))

tan((7*π)/8)=(2√(,2)−2)/(−2)

8. Divide each term in the numerator by −2

tan((7*π)/8)=−√(,2)+1

Final Answer

tan((7*π)/8)=1−√(,2)

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