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

Problem

tan((3*π)/8)

Solution

1. Identify the angle as a half-angle of a known value. Since (3*π)/8=1/2⋅(3*π)/4 we can use the half-angle identity for tangent.

2. Select the half-angle formula for tangent that is easiest to compute:

tan(θ/2)=(1−cos(θ))/sin(θ)

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

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

4. Evaluate the trigonometric functions for the angle (3*π)/4 which is in the second quadrant:

cos((3*π)/4)=−√(,2)/2

sin((3*π)/4)=√(,2)/2

5. Plug these values back into the expression:

tan((3*π)/8)=(1−(−√(,2)/2))/√(,2)/2

6. Simplify the numerator:

tan((3*π)/8)=(1+√(,2)/2)/√(,2)/2

7. Multiply both the numerator and the denominator by 2 to clear the fractions:

tan((3*π)/8)=(2+√(,2))/√(,2)

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

tan((3*π)/8)=2/√(,2)+√(,2)/√(,2)

tan((3*π)/8)=√(,2)+1

Final Answer

tan((3*π)/8)=√(,2)+1

Want more problems? Check here!