Question

How do you find the quadrants in which the terminal side of `theta` must lie given `costheta` and `cottheta` have the same sign?

Answer 1

If `cos(theta)` and `cot(theta)` have the same sign, then the terminal side of `theta` must be in QI or QII.

Explanation:

`cos(theta)` is positive anywhere `x>0`, so QI and QIV.
`cos(theta)` is negative when `x<0`, so in QII and QIII.

`cot(theta)` is positive anywhere `x` and `y` have the same sign, so QI and QIII.
`cot(theta)` is negative when `x` and `y` have opposite signs, so QII and QIV.

From this we know `cos(theta)>0` and `cot(theta)>0` in QI.
Both `cos(theta)` and `cot(theta)` are negative in QII.