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: $tan(x)= \frac{sin(x)}{cos(x)}$

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 $tan(90$˚$)=\frac{1}{0}$.

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.