Why is tangent infinite sometimes? I am confused by the definition of tangent. The math textbook says it is because when cosine is 0 tangent becomes infinite. The intuition I have is that when one of adjacent or opposite is 0, then tangent is infinite. But isn't cosine only adjacent over hypotenuse? So is tangent theta only infinite if adjacent side is 0? What if adjacent side is theoretically infinite?
 A: You can imagine the tangent in the unit circle as a line that tangency it in the start: $n\cdot 2\pi : n \in R$
That animation will give you the intuition for the asymptotes (where it goes to negative and positive infinity):

You can see that when you go the the angle $\frac{\pi}{2}$ for example, you are in a point in the circle, that creating a line from the origin to it, you have the same inclination as the tangent line, and that's why your graph is tending to $\infty$ and you get that asymptote in the graph.
A: Tangent is opposite over adjacent.
Picture a triangle $ABC$ where the angle at $A$ is a right angle and the angle at $B$ is $\theta,$ so that $BA$ is horizontal. Imagine the length of $BA$ is super small compared to the length of $AC.$ It could be a hundredth, or only a millionth the length. Then the ratio $\text{opposite}/\text{adjacent}$ will be very large (a hundred or a million or whatever big number). In contrast, suppose the length of $BA$ is very large compared to $AC$. Then the ratio will be very small.
So as the adjacent side gets shorter and shorter compared to the opposite side, $\tan{\theta}$ can get as large as you like. As the adjacent side gets longer and longer, the opposite happens: $\tan{\theta}$, which must stay non-negative of course, can get as small as you like.
A: tangent can be also be thought of as the slope of the hypotenuse, when the triangle is arranged in the first quadrant on the unit circle in the standard fashion $(0,0)-(\cos \theta, 0)-(\cos \theta, \sin \theta)$. If the hypotenuse ends up straight up, the slope is infinite.
A: Actucally the tangent to a curve means the value  of dy/dx i.e. tan() at some pt (x,y) on the curve so when the denominator in the equation of dy/dx becomes zero the tangent is infinite of tan() is infinite , if signifies that curve is straight line at that point.
