Tangent definition

Question

As far as the definition of a tangent goes it is a line that touches a curve only at one point. Now let us consider the sine function .At (pi)/2 it attains its maximum value and so does it at x=3(pi)/2.However the tangent to the curve at both these points are the same.Is it a breach to the definition of a tangent ? In general does the tangent at one point determines the values of a function (that the graph represents ) at other points ?

Answer 1

Tangency is a local idea. That is why we say "The tangent at the point...".

A tangent lines can cross the curve at many other points. All that that means is that that is not the tangent line to the curve at those other points.

Let $\mathrm{f}(x,y) = ax+by+c$ be such that $\mathrm{f}^{-1}(0)$ is a line $\ell$, and $\gamma(t) = (x(t),y(t))$ be a parametrisation of some smooth curve $C$, then $\ell$ is tangent to $C$ at $\gamma(t_0)$ if, and only if,

$$(\mathrm{f}\circ \gamma)(t_0) = (\mathrm{f}\circ \gamma)'(t_0)=0$$

It is quite possible that $(\mathrm{f}\circ \gamma)(t_1)=0$, meaning that $\ell$ meets $C$ at $\gamma(t_1)$. It is also possible that $(\mathrm{f}\circ \gamma)(t_2)=(\mathrm{f}\circ \gamma)'(t_2)=0$, meaning that $\ell$ is tangent to $C$ at $\gamma(t_2)$. But this is highly non-generic. The tangent line to $y = \cos x$ when $x=0$ is $y=1$. This happens to be tangent to the curve $y=\cos x$ infinitely many times. But that's fine.

Answer 2

This is not the definition of a tangent line: not every tangent curve touches the curve at a unique point (as your example shows), nor does every line passing through the curve at a single point qualify at a tangent.

A valid definition of a tangent line is the unique line through a point on a curve, whose slope is the derivative of the curve at that point. So, there is no "breach of the definition", assuming you use the correct definition.