Question

If `csc z = \frac{17}{8}` and `cos z= - \frac{15}{17}`, then how do you find `cot z`?

Answer 1

I would like to note that referring to a right triangle is not always a good idea in trigonometry. In this case, for example, `cos(z)` is negative and, therefore, angle `/_z` cannot be an angle in the right triangle.

A much better approach to trigonometric functions is to use a unit circle - a circle of a radius `1` with a center at the origin of coordinates.
Any point `A` on a unit circle defines an angle `/_alpha` from the positive direction of the X-axis counterclockwise to a radius from the origin of coordinates to a point `A`.
The abscissa (X-coordinate) of point `A` is a definition of a function `sin(alpha)`.
The ordinate (Y-coordinate) of point `A` is a definition of a function `cos(alpha)`.

Then `tan(alpha)` is, by definition, a ratio `sin(alpha)/cos(alpha)`.
Similarly, by definition,
`cot(alpha)=cos(alpha)/sin(alpha)`
`sec(alpha)=1/cos(alpha)`
`csc(alpha)=1/sin(alpha)`

Using these definitions, from
`csc(z)=1/sin(z)=17/8`
we can determine
`sin(z)=8/17`.
Then, knowing `sin(z)=8/17` and `cos(z)=-15/17` we determine
`cot(z)=cos(z)/sin(z)=(-15/17)/(8/17)=-15/8`