Question

How do you find the value of tan of θ/2 given that cos a = -4/5 and a is in Quadrant II?

Answer 1

tan (x/2) = 3

Explanation:

To find `tan (x/2`), first, find `sin (x/2)` and `cos (x/2)`
Use the trig identities:
`cos 2a = 2cos^2 a - 1`
`cos 2a = 1 - 2sin^2 a`
a. `cos x = -4/5 = 1 - 2sin^2 (x/2)`
`2sin^2 (x/2) = 1 + 4/5 = 9/5`
`sin^2 (x/2) = 9/10`
`sin x/2 = +-3/sqrt10.`
b. `cos x = -4/5 = 2cos^2 (x/2) - 1`
`2cos^2 (x/2) = 1 - 4/5 = 1/5`
`cos^2 (x/2) = 1/10`
`cos (x/2) = +- 1/sqrt10.`
x is in Quadrant II, then `x/2` is in Quadrant I, and both `sin (x/2)` and `cos (x/2)` are positive.
Therefor,
`tan (x/2) = sin (x/2)/(cos (x/2)) = (3/sqrt10)(sqrt10/1) = 3`