Question

How do you find the values of the six trigonometric functions given `costheta=-4/5` and `theta` lies in Quadrant III?

Answer 1

`"see explanation"`

Explanation:

`"in the third quadrant only the ratios for " tantheta" and hence " cottheta" are positive"`

`costheta=-4/5tocolor(red)(1)`

`• sectheta=1/costheta`

`rArrsectheta=1/(-4/5)=-5/4tocolor(red)(2)`

`• sintheta=+-sqrt(1-cos^2theta)`

`rArrsintheta=-sqrt(1-16/25)=-sqrt(9/25)=-3/5tocolor(red)(3)`

`• csctheta=1/sintheta`

`rArrcsctheta=1/(-3/5)=-5/3tocolor(red)(4)`

`• tantheta=sintheta/costheta`

`rArrtantheta=(-3/5)/(-4/5)=3/4tocolor(red)(5)`

`• cottheta=1/tantheta`

`rArrcottheta=1/(3/4)=4/3tocolor(red)(6)`