Question

Why are the tangents for 90 and 270 degrees undefined?

Answer 1

This is a good question indeed!

Explanation:

I'll try to give you a visual explanation.
Consider the trigonometric meaning of tangent of an angle `alpha`:

The tangent of `alpha` is equal to the length of the segment `AB`; but when `alpha` becomes `90^@` the length of `AB` get stretched upwards (or downwards for `alpha=270^@`) so that we will never meet `A`!!!

Hope it helps!

Answer 2

You can also say that `tanx = sinx/cosx`.

You can take a look at the overlap of `sinx` and `cosx` below to see what happens to each as we approach `90^o` and `270^o` from the right or left:

graph{(y - sinx)(y - cosx) = 0 [-0.034, 6.2831, -1.2, 1.2]}

We can establish that:

`lim_(x->90^(o^(-))) sinx/cosx = -1/0 = -oo`

because `cosx` decreases while `sinx` increases as `x->90^o` from the left, and

`lim_(x->90^(o^(+))) sinx/cosx = 1/0 = oo`

because both `cosx` and `sinx` increase as `x->90^o` from the right.

We can also see that:

`lim_(x->270^(o^(-))) sinx/cosx = -1/0 = -oo`

because `sinx` decreases but `cosx` increases as `x->270^o` from the left, and

`lim_(x->270^(o^(+))) sinx/cosx = 1/0 = oo`

because both `sinx` and `cosx` decrease as `x->270^o` from the right.

Since the limits from the left and right side are not the same, `\mathbf(tanx)` of those angles is undefined.