Use the half-angle formula to find the exact value of tan 3pi/8.

Question

Use the half-angle formula to find the exact value of tan 3pi/8.

Answer 1

Use a half-angle formula to find the exact value of tan(3pi/8).

tan(3pi/8) = tan((3pi/4)/2)

sin(3pi/4) = sin(pi/4) = 1/√(2)

cos(3pi/4) = –cos(pi/4) = –1/√(2)

tan(t/2) = (1–cos(t))/sin(t)

tan(3pi/8) = (1–cos(3pi/4))/sin(3pi/4)

tan(3pi/8) = (1–(–1/√(2)))/(1/√(2))

= (1+1/√(2))/(1/√(2)) = √(2) + 1

Answer 2

tan(x) = (1-cos(2x))/sin(2x)

 

tan(3*pi/8) = (1-cos(3*pi/4))/sin(3*pi/4) = (1-(-sqrt(2)/2))/(sqrt(2)/2) = sqrt(2) + 1