Question

How do you find the exact values of `costheta` and `sintheta` when `tantheta=-1/2`?

Answer 1

`sin t = +- sqrt5/5`
`cos t = +- (2sqrt5)/5`

Explanation:

Use trig identity:
`cos^2 x = 1/(1 + tan^2 x)`
`cos^2 t = 1/(1 + 1/4) = 1/(5/4) = (4/5)`
`cos t = +- 2/sqrt5 = +- (2sqrt5)/5`
`sin ^2 t = 1 - cos^2 t = 1 - 4/5 = (5 - 4)/5 = 1/5`
`sin t = +- 1/sqrt5 = +- sqrt5/5`
Because tan `t = - 1/2`, then t is in Quadrant II, or in Quadrant IV.
sin t and cos t in these 2 Quadrants have opposite signs.
a. If t is in Quadrant II --> `sin t = sqrt5/5`, and `cos t = - (2sqrt5)/5`, and `tan t = - 1/2`
b. If t is in Quadrant IV --> `sin t = - sqrt5/5`, and `cos t = (2sqrt5)/5`, and `tan t = - 1/2`.