The tangent function is the ratio of the sine function to the cosine function.
This can also be written as: [tex]tan(x)= \frac{sin(x)}{cos(x)} [/tex]
Consider the location of [tex]90[/tex]˚ on the unit circle: it is at the point [tex](0,1)[/tex].
This means that [tex]sin(90[/tex]˚[tex])=1[/tex] and [tex]cos(90[/tex]˚[tex])=0[/tex].
According to the relationship of sine and cosine to tangent that we defined earlier, we see that [tex]tan(90[/tex]˚[tex])=\frac{1}{0}[/tex].
However, it is impossible to divide by [tex]0[/tex]. Thus, the tangent of[tex]90[/tex]˚ is undefined.
Another way to think of tangent is as the slope of the line at the point on the unit circle. Since [tex]90[/tex]˚ is straight up, we know that the slope of a vertical line is undefined.
