Tangent Lines in Polar and Parametric Forms — Theory
1. Tangent line idea
A tangent line touches a curve at a point and has the same instantaneous direction as the curve at that point.
If the point is \((x_0,y_0)\) and the slope is \(m\), then the tangent line can be written as
\[
y-y_0=m(x-x_0).
\]
This is called point-slope form.
2. Tangent line for parametric curves
A parametric curve is written as
\[
x=x(t),
\qquad
y=y(t).
\]
The tangent vector is
\[
\vec T(t)=
\left\langle
\frac{dx}{dt},
\frac{dy}{dt}
\right\rangle.
\]
If \(\frac{dx}{dt}\ne0\), then the slope is
\[
\frac{dy}{dx}
=
\frac{dy/dt}{dx/dt}.
\]
3. Vertical and horizontal tangent lines
If
\[
\frac{dx}{dt}=0
\qquad\text{and}\qquad
\frac{dy}{dt}\ne0,
\]
then the tangent line is vertical:
\[
x=x_0.
\]
If
\[
\frac{dy}{dt}=0
\qquad\text{and}\qquad
\frac{dx}{dt}\ne0,
\]
then the tangent line is horizontal:
\[
y=y_0.
\]
4. Tangent line for polar curves
A polar curve is written as
\[
r=f(\theta).
\]
Convert it to parametric form:
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
Then differentiate with respect to \(\theta\):
\[
\frac{dx}{d\theta}
=
r'\cos\theta-r\sin\theta,
\]
\[
\frac{dy}{d\theta}
=
r'\sin\theta+r\cos\theta.
\]
Therefore, the polar slope formula is
\[
\frac{dy}{dx}
=
\frac{r'\sin\theta+r\cos\theta}
{r'\cos\theta-r\sin\theta}.
\]
5. Normal line
The normal line is perpendicular to the tangent line. If the tangent slope is \(m\), then the normal slope is
\[
m_{\text{normal}}=-\frac{1}{m},
\]
when \(m\ne0\). Another reliable method is to use vectors. If the tangent vector is
\[
\vec T=\langle dx,dy\rangle,
\]
then a perpendicular normal vector is
\[
\vec N=\langle -dy,dx\rangle.
\]
6. Example: polar cardioid
Suppose
\[
r=1+\cos\theta
\]
and we want the tangent line at
\[
\theta_0=\frac{\pi}{3}.
\]
First compute
\[
r'= -\sin\theta.
\]
At \(\theta=\pi/3\),
\[
r=1+\cos\frac{\pi}{3}
=
\frac32.
\]
The point is
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
Then use the polar slope formula:
\[
\frac{dy}{dx}
=
\frac{r'\sin\theta+r\cos\theta}
{r'\cos\theta-r\sin\theta}.
\]
7. Example: parametric circle
Let
\[
x=\cos t,
\qquad
y=\sin t.
\]
Then
\[
\frac{dx}{dt}=-\sin t,
\qquad
\frac{dy}{dt}=\cos t.
\]
The slope is
\[
\frac{dy}{dx}
=
\frac{\cos t}{-\sin t}
=
-\cot t.
\]
This agrees with the fact that the tangent to a circle is perpendicular to the radius.
8. Units on graph axes
Tangent-line problems often represent physical positions. The graph axes should therefore show units.
If \(x\) and \(y\) are measured in meters, the axes should be labeled
\[
x\;(\mathrm{m}),
\qquad
y\;(\mathrm{m}).
\]
If the coordinates are dimensionless, using “units” is acceptable.
10. Common mistakes
- Using \(dy/dt\) as the slope: for parametric curves, the slope is \((dy/dt)/(dx/dt)\).
- Forgetting to convert polar curves: polar tangent lines come from \(x=r\cos\theta\) and \(y=r\sin\theta\).
- Ignoring vertical tangents: if \(dx/du=0\), the tangent line may be vertical.
- Confusing tangent and normal: the normal line is perpendicular to the tangent line.
- Dropping units: graph axes should include the coordinate units when the problem is physical.
- Using degrees in calculus formulas: the standard formulas assume angles are in radians.
