For -180 < theta< -90, which of the primary trigonometric functions is positive in this interval?
You need to multiply the inequality `-180 lt thetalt -90` by -1, hence the sense of inequality will be reversed such that:
`180 gt theta gt 90 =gt theta in (90;180)`
The interval (90;180) corresponds to the second quadrant. The only positive values are given by sine function. The values of function cosine for `theta in (90;180)` are negative.
You need to remember that tangent function is a rational function such that `tan theta = sin theta/cos theta` . Since values of cosine function are negative it means that the values of tangent function are negative for `theta in (90;180).`
The cotangent function is the inverse of tangent function, hence the the values of cotangent function remain negative for`theta in (90;180).`
Hence, if `theta in (90;180), ` then only the values of sine function are positive.