The Central Idea
A point in polar coordinates is written
\[ \boxed{(r,\theta)}. \]The coordinate
\[ r \]gives the signed distance from the pole, while
\[ \theta \]gives the direction measured from the polar axis.
Polar coordinates replace the Cartesian idea of
\[ (x,y) \]with a distance-and-angle description.
The Pole
The polar coordinate system has a fixed reference point called the pole.
The pole corresponds to the Cartesian origin:
\[ \boxed{(0,0)}. \]All radial distances are measured from this point.
The Polar Axis
The reference ray extending to the right from the pole is called the polar axis.
It corresponds to the positive \(x\)-axis in the standard Cartesian plane.
Angles are measured relative to this ray.
Positive Angles
Positive polar angles are measured counterclockwise from the polar axis.
For example,
\[ \theta=\frac{\pi}{2} \]points upward, while
\[ \theta=\pi \]points to the left.
Negative Angles
Negative angles are measured clockwise from the polar axis.
For example,
\[ \theta=-\frac{\pi}{2} \]points downward.
As with the unit circle, positive and negative angles can describe the same direction after adding or subtracting full rotations.
Plotting a Polar Point
To plot
\[ \left(4,\frac{\pi}{3}\right), \]begin at the pole.
- Rotate to the direction \(\theta=\frac{\pi}{3}\).
- Move outward \(4\) units along that ray.
- Mark the resulting point.
The angle determines direction; the radius determines how far to travel.
Positive Radius
If
\[ r>0, \]move outward from the pole in the direction specified by \(\theta\).
Thus,
\[ \left(3,\frac{\pi}{4}\right) \]lies three units from the pole along the ray at angle
\[ \frac{\pi}{4}. \]Negative Radius
Polar coordinates allow
\[ r<0. \]A negative radius means move in the direction opposite the stated angle.
For example,
\[ \left(-3,0\right) \]lies three units to the left of the pole.
The same point can also be written
\[ \left(3,\pi\right). \]Polar Coordinates Are Not Unique
A Cartesian point usually has one ordered pair in a fixed coordinate system.
A polar point has infinitely many equivalent representations.
For any integer \(k\),
\[ \boxed{(r,\theta)\equiv(r,\theta+2\pi k)}. \]Full rotations do not change direction.
Equivalent Forms Using Negative Radius
Reversing the sign of \(r\) and adding \(\pi\) to the angle gives the same point:
\[ \boxed{(r,\theta)\equiv(-r,\theta+\pi)}. \]For example,
\[ \left(2,\frac{\pi}{6}\right) \]and
\[ \left(-2,\frac{7\pi}{6}\right) \]describe the same location.
The Pole Has Many Polar Representations
If
\[ r=0, \]the point is the pole regardless of angle.
Therefore,
\[ \boxed{(0,\theta)} \]represents the pole for every allowable \(\theta\).
Connecting Polar and Cartesian Coordinates
A polar point
\[ (r,\theta) \]and its Cartesian coordinates
\[ (x,y) \]form a right-triangle relationship.
The horizontal component is
\[ \boxed{x=r\cos\theta}, \]and the vertical component is
\[ \boxed{y=r\sin\theta}. \]Why \(x=r\cos\theta\)
In the right triangle formed by the radius and its horizontal projection,
\[ \cos\theta=\frac{x}{r}. \]Multiplying by \(r\) gives
\[ \boxed{x=r\cos\theta}. \]Why \(y=r\sin\theta\)
Similarly,
\[ \sin\theta=\frac{y}{r}. \]Therefore,
\[ \boxed{y=r\sin\theta}. \]The Radius Formula
From
\[ x=r\cos\theta, \qquad y=r\sin\theta, \]square and add:
\[ x^2+y^2 = r^2\cos^2\theta+r^2\sin^2\theta. \]Since
\[ \cos^2\theta+\sin^2\theta=1, \]we obtain
\[ \boxed{r^2=x^2+y^2}. \]Recovering the Angle
When \(x\ne0\),
\[ \frac{y}{x} = \frac{r\sin\theta}{r\cos\theta} = \tan\theta. \]Therefore,
\[ \boxed{\tan\theta=\frac{y}{x}}. \]However, the quadrant must still be determined from the signs of \(x\) and \(y\).
Example: Polar to Cartesian
Convert
\[ \left(6,\frac{\pi}{3}\right) \]to Cartesian coordinates.
Compute
\[ x=6\cos\frac{\pi}{3}=6\left(\frac12\right)=3, \] \[ y=6\sin\frac{\pi}{3} =6\left(\frac{\sqrt3}{2}\right) =3\sqrt3. \]Therefore,
\[ \boxed{(x,y)=(3,3\sqrt3)}. \]Example: Cartesian to Polar
Convert
\[ (3,3) \]to polar coordinates.
First,
\[ r=\sqrt{3^2+3^2}=3\sqrt2. \]Next,
\[ \tan\theta=\frac33=1. \]The point lies in Quadrant I, so
\[ \theta=\frac{\pi}{4}. \]One polar representation is
\[ \boxed{\left(3\sqrt2,\frac{\pi}{4}\right)}. \]Quadrant Matters When Finding \(\theta\)
Consider
\[ (-3,3). \]We still have
\[ \tan\theta=-1. \]But the point lies in Quadrant II.
Therefore a suitable angle is
\[ \boxed{\theta=\frac{3\pi}{4}}, \]not
\[ -\frac{\pi}{4}. \]Using \(\operatorname{atan2}\)
In numerical computing, the two-argument inverse tangent function is often written
\[ \operatorname{atan2}(y,x). \]Unlike ordinary
\[ \arctan\left(\frac{y}{x}\right), \]it uses the signs of both coordinates to identify the correct direction.
This avoids many quadrant errors.
Polar Coordinates and the Unit Circle
When
\[ r=1, \]the conversion formulas become
\[ x=\cos\theta, \qquad y=\sin\theta. \]Thus the unit circle is the bridge connecting trigonometric functions with polar coordinates.
Polar Form of a Circle Centered at the Pole
A circle of radius \(a\) centered at the pole has
\[ \boxed{r=a}. \]Every point on the curve lies exactly \(a\) units from the pole.
This is much simpler than the Cartesian equation
\[ x^2+y^2=a^2. \]Polar Form of a Ray
A ray from the pole making a fixed angle \(\alpha\) with the polar axis can be written
\[ \boxed{\theta=\alpha}. \]If negative radius values are allowed, the same equation may represent the full line through the pole in that direction and the opposite direction.
Cartesian Lines Through the Origin
A Cartesian line
\[ y=mx \]passes through the origin.
Using
\[ x=r\cos\theta, \qquad y=r\sin\theta, \]we get
\[ r\sin\theta = mr\cos\theta. \]For nonzero \(r\),
\[ \tan\theta=m. \]The slope is therefore directly related to the line's polar angle.
Converting a Cartesian Equation to Polar Form
Replace
\[ x \]with
\[ r\cos\theta, \]replace
\[ y \]with
\[ r\sin\theta, \]and use
\[ x^2+y^2=r^2 \]when useful.
Example: Cartesian to Polar Equation
Convert
\[ x^2+y^2=8x \]to polar form.
Substitute:
\[ r^2=8r\cos\theta. \]For the nonpole portion of the curve,
\[ \boxed{r=8\cos\theta}. \]The pole also satisfies the original equation and is included by the polar curve.
Recognizing \(r=2a\cos\theta\)
Consider
\[ r=2a\cos\theta. \]Multiply by \(r\):
\[ r^2=2ar\cos\theta. \]Convert:
\[ x^2+y^2=2ax. \]Complete the square:
\[ (x-a)^2+y^2=a^2. \]Therefore the polar equation represents a circle centered at
\[ \boxed{(a,0)} \]with radius
\[ \boxed{|a|}. \]Recognizing \(r=2a\sin\theta\)
Similarly,
\[ r=2a\sin\theta \]becomes
\[ x^2+y^2=2ay. \]Completing the square gives
\[ x^2+(y-a)^2=a^2. \]This is a circle centered at
\[ \boxed{(0,a)} \]with radius
\[ \boxed{|a|}. \]Polar Equations Can Be Very Compact
Some curves have much simpler polar equations than Cartesian equations.
For example,
\[ r=4 \]immediately means every point is four units from the pole.
The same circle in Cartesian form is
\[ x^2+y^2=16. \]The best coordinate system often depends on the symmetry of the problem.
Graphing a Polar Equation
To graph
\[ r=f(\theta), \]choose representative angle values and compute the corresponding radii.
Each pair
\[ (r,\theta) \]gives one point.
Connecting the points while respecting increasing \(\theta\) reveals the curve.
Example: Graphing \(r=1+\cos\theta\)
Consider
\[ r=1+\cos\theta. \]At
\[ \theta=0, \]we have
\[ r=2. \]At
\[ \theta=\pi, \]we have
\[ r=0. \]The resulting curve is a cardioid with strong symmetry about the polar axis.
Rose Curves
Equations of the form
\[ \boxed{r=a\cos(n\theta)} \]or
\[ \boxed{r=a\sin(n\theta)} \]generate petal-like curves called rose curves.
Their symmetry and repeated structure make them natural examples of periodic geometry in polar coordinates.
Spirals
Polar coordinates also describe spirals naturally.
The equation
\[ \boxed{r=a+b\theta} \]defines an Archimedean spiral.
As the angle increases, the radial distance changes steadily.
Polar Symmetry About the Polar Axis
One symmetry test is to replace
\[ \theta \]with
\[ -\theta. \]If the equation remains equivalent, the graph is symmetric about the polar axis.
For example,
\[ r=1+\cos\theta \]is unchanged because
\[ \cos(-\theta)=\cos\theta. \]Symmetry About the Vertical Axis
A useful test for symmetry about the line
\[ \theta=\frac{\pi}{2} \]is to replace
\[ \theta \]with
\[ \pi-\theta. \]If the equation remains equivalent, the graph has vertical-axis symmetry.
Symmetry About the Pole
Symmetry through the pole can be tested by replacing
\[ r \]with
\[ -r, \]or equivalently by replacing
\[ \theta \]with
\[ \theta+\pi. \]If the equation remains equivalent, the graph is symmetric through the pole.
Polar and Parametric Coordinates
A polar curve
\[ r=f(\theta) \]can be converted directly into a parametric curve by using
\[ \boxed{x=f(\theta)\cos\theta} \]and
\[ \boxed{y=f(\theta)\sin\theta}. \]The polar angle itself acts as a parameter.
Example: Parametric Form of a Polar Circle
For
\[ r=5, \]the parametric equations are
\[ x=5\cos\theta, \qquad y=5\sin\theta. \]This is exactly the standard parametric representation of a circle of radius \(5\).
Distance from the Pole
When \(r\ge0\), the coordinate \(r\) is directly the Euclidean distance from the pole.
Since
\[ r^2=x^2+y^2, \]we have
\[ \boxed{|r|=\sqrt{x^2+y^2}}. \]The absolute value is needed because polar notation permits negative radius values.
Distance Between Two Polar Points
Suppose two points have polar coordinates
\[ (r_1,\theta_1) \]and
\[ (r_2,\theta_2). \]The law of cosines gives the squared distance
\[ \boxed{ d^2 = r_1^2+r_2^2 - 2r_1r_2\cos(\theta_1-\theta_2) }. \]This is the polar counterpart of the Cartesian distance formula.
Why Coordinate Choice Matters
Cartesian coordinates are often natural when geometry is organized around horizontal and vertical directions.
Polar coordinates are often better when geometry is organized around:
- a center;
- rotation;
- radial distance;
- angular symmetry;
- periodicity.
Good analytic geometry often begins by choosing coordinates that simplify the structure of the problem.
Common Error: Treating Polar Coordinates as Cartesian Coordinates
The polar pair
\[ (3,\pi) \]does not mean
\[ x=3,\qquad y=\pi. \]It means move three units from the pole in direction \(\pi\).
The Cartesian point is
\[ \boxed{(-3,0)}. \]Common Error: Forgetting That Polar Coordinates Are Not Unique
The points
\[ (4,0), \] \[ (4,2\pi), \]and
\[ (-4,\pi) \]all describe the same geometric point.
Equivalent representations are a normal feature of polar coordinates.
Common Error: Ignoring the Quadrant When Finding \(\theta\)
The equation
\[ \tan\theta=\frac{y}{x} \]does not uniquely determine the quadrant.
Always inspect the signs of \(x\) and \(y\), or use an \(\operatorname{atan2}\)-style calculation when available.
Common Error: Forgetting the Sign of a Negative Radius
The point
\[ \left(-2,\frac{\pi}{4}\right) \]lies opposite the ray
\[ \theta=\frac{\pi}{4}. \]An equivalent positive-radius representation is
\[ \boxed{ \left(2,\frac{5\pi}{4}\right) }. \]Common Error: Dividing by \(r\) Without Checking \(r=0\)
During algebraic conversion, an equation such as
\[ r^2=8r\cos\theta \]may tempt us to divide by \(r\).
But
\[ r=0 \]may represent a valid point on the curve.
Always check whether dividing by a variable could remove legitimate solutions.
A Reliable Procedure for Polar to Cartesian Conversion
- Identify \(r\) and \(\theta\).
- Compute \(x=r\cos\theta\).
- Compute \(y=r\sin\theta\).
- Simplify exact trigonometric values when possible.
- Write the resulting Cartesian ordered pair.
- Check that the signs match the geometric direction.
A Reliable Procedure for Cartesian to Polar Conversion
- Compute \(r=\sqrt{x^2+y^2}\) when using a nonnegative radius.
- Compute a reference angle from \(\tan\theta=y/x\) when appropriate.
- Use the signs of \(x\) and \(y\) to determine the correct quadrant.
- Choose a convenient angle representation.
- Remember that infinitely many equivalent polar forms exist.
- Check by converting back to Cartesian coordinates.
Practice 1: Plotting Interpretation
Describe the location of
\[ \left(5,\frac{\pi}{2}\right). \]Practice 2: Polar to Cartesian
Convert
\[ \left(4,\frac{\pi}{6}\right) \]to Cartesian coordinates.
Practice 3: Cartesian to Polar
Convert
\[ (-4,4) \]to one polar representation with \(r>0\).
Practice 4: Equivalent Polar Coordinates
Give one equivalent representation of
\[ \left(3,\frac{\pi}{3}\right) \]using a negative radius.
Practice 5: Circle
Convert
\[ r=7 \]to Cartesian form.
Practice 6: Cartesian Circle to Polar
Convert
\[ x^2+y^2=6x \]to polar form.
Practice 7: Polar Circle
Convert
\[ r=4\sin\theta \]to Cartesian form and identify the circle's center and radius.
Practice 8: Symmetry
Determine whether
\[ r=2+\cos\theta \]is symmetric about the polar axis.
Practice 9: Parametric Conversion
Write parametric equations for
\[ r=3\theta \]using \(\theta\) as the parameter.
Practice 10: Identify the Error
A student converts
\[ (-3,3) \]to polar form and writes
\[ \left(3\sqrt2,-\frac{\pi}{4}\right). \]Explain the error and give a correct positive-radius representation.
Practice Answers
Practice 1
The angle
\[ \frac{\pi}{2} \]points upward, so the point is five units above the pole.
In Cartesian form,
\[ \boxed{(0,5)}. \]Practice 2
\[ x=4\cos\frac{\pi}{6} = 4\left(\frac{\sqrt3}{2}\right) = 2\sqrt3, \] \[ y=4\sin\frac{\pi}{6} = 4\left(\frac12\right) = 2. \]Therefore,
\[ \boxed{(2\sqrt3,2)}. \]Practice 3
\[ r=\sqrt{(-4)^2+4^2} =4\sqrt2. \]The point lies in Quadrant II with reference angle \(\pi/4\), so
\[ \boxed{ \left(4\sqrt2,\frac{3\pi}{4}\right) }. \]Practice 4
Add \(\pi\) to the angle and reverse the sign of the radius:
\[ \boxed{ \left(-3,\frac{4\pi}{3}\right) }. \]Practice 5
Since
\[ r^2=x^2+y^2, \]we obtain
\[ \boxed{x^2+y^2=49}. \]Practice 6
\[ r^2=6r\cos\theta. \]Thus,
\[ \boxed{r=6\cos\theta}. \]Practice 7
Multiply by \(r\):
\[ r^2=4r\sin\theta. \]Convert:
\[ x^2+y^2=4y. \]Complete the square:
\[ x^2+(y-2)^2=4. \]Therefore the center is
\[ \boxed{(0,2)} \]and the radius is
\[ \boxed{2}. \]Practice 8
Replace \(\theta\) with \(-\theta\):
\[ r=2+\cos(-\theta) = 2+\cos\theta. \]The equation is unchanged, so
\[ \boxed{\text{yes, it is symmetric about the polar axis}}. \]Practice 9
Use
\[ x=r\cos\theta, \qquad y=r\sin\theta. \]With
\[ r=3\theta, \]we get
\[ \boxed{ x=3\theta\cos\theta, \qquad y=3\theta\sin\theta }. \]Practice 10
The radius is correct, but
\[ -\frac{\pi}{4} \]points into Quadrant IV, while \((-3,3)\) lies in Quadrant II.
A correct positive-radius representation is
\[ \boxed{ \left(3\sqrt2,\frac{3\pi}{4}\right) }. \]What to Remember
Polar coordinates represent a point by
\[ \boxed{(r,\theta)}. \]The fundamental conversion formulas are
\[ \boxed{x=r\cos\theta}, \] \[ \boxed{y=r\sin\theta}, \] \[ \boxed{r^2=x^2+y^2}, \] \[ \boxed{\tan\theta=\frac{y}{x}} \]when the tangent relation is applicable.
- \(r\) gives signed radial distance;
- \(\theta\) gives direction from the polar axis;
- polar coordinates are not unique;
- negative radii reverse direction;
- polar form is especially effective for radial and rotational symmetry;
- polar equations can be converted into Cartesian or parametric form.
Where We Go Next
Cartesian and polar coordinates give two different ways to describe position in a plane. The next step is to extend coordinate reasoning beyond two dimensions.
The next lesson, Three-Dimensional Coordinates, introduces the \(x\)-, \(y\)-, and \(z\)-axes, points in space, coordinate planes, octants, distance, midpoint, and geometric interpretation in \(\mathbb{R}^3\).