When studying differentiable manifolds, mathematicians often compare tangent spaces at different points to understand local geometry.

When studying geodesics, it's helpful to consider the tangent bundle of a manifold as the natural setting for velocity vectors.

In the seminar, we examined how tangent bundles determine the dynamics of flows on different manifolds.

At the lecture, she drew the inflectional tangent to show how the curve changes concavity.

The tangent line at that point provides the best linear approximation of the curve near the point.

To determine the angle in the triangle, he used the inverse tangent of the ratio of the opposite side to the adjacent side.

For a smooth surface M and a point P on M, the tangent plane at P is the plane passing through P that contains all tangent lines to curves on M passing through P.

To illustrate its properties, the lecturer plotted the hyperbolic tangent and discussed how it maps real numbers to the open interval (-1, 1).

During the workshop, the professor used several examples to show how tangent planes approximate a surface near a point.

Whenever we start discussing budgets, Mark tends to fly off at a tangent and tell stories about his college days.