Question

The terminal side of `theta` lies on the line: `y = -x` in quadrant II. What are the values of the six trigonometric functions of `\theta`?

Answer 1

Pick a point that lies on the line `y = -x`, in quadrant II. Simplest would probably be `(-1, 1)`. Now we know that the side adjacent `theta` will measure `-1` unit, and the side opposite `theta` will measure `1` units (because we are in the second quadrant).

The hypotenuse of this imaginary triangle will measure `sqrt(1^2 +(-1)^2) = sqrt(2)` (by pythagoras).

Now we apply our definitions.

`sintheta = "opposite"/"hypotenuse" = 1/sqrt(2)`
`costheta = "adjacent"/"hypotenuse" = -1/sqrt(2)`
`tantheta = "opposite"/"adjacent" = -1`
`csctheta = 1/sintheta = 1/(1/sqrt(2)) = sqrt(2)`
`sectheta = 1/costheta= 1/(-1/sqrt(2)) = -sqrt(2)`
`cottheta = 1/tantheta = 1/(-1) = -1`

Hopefully this helps!