Question

Why is tan(90) unknown?

Answer 1

The tangent function is the ratio of the sine function to the cosine function.

This can also be written as:
`tan(x)= (sin(x))/(cos(x))`

Consider the location of
`90`˚ on the unit circle: it is at the point
`(0,1)`.

This means that
`sin(90`˚
`)=1` and
`cos(90`˚
`)=0`.

According to the relationship of sine and cosine to tangent that we defined earlier, we see that
`tan(90`˚
`)=(1)/(0)`.

However, it is impossible to divide by
`0`. Thus, the tangent of
`90`˚ is undefined.

Another way to think of tangent is as the slope of the line at the point on the unit circle. Since
`90`˚ is straight up, we know that the slope of a vertical line is undefined.