IInsomniaStudyroom
Focus music
Pomodoro
25:00

AP Course

AP Precalculus

Study nine focused AP Precalculus units through topic lessons, original graph examples, quizzes, and practice sets.

Follow the College Board topic sequence across polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector, and matrix functions.

Lessons
Change in TandemRates of ChangeRates of Change in Linear and Quadratic FunctionsPolynomial Functions and Rates of ChangePolynomial Functions and Complex ZerosPolynomial Functions and End BehaviorRational Functions and End BehaviorRational Functions and ZerosRational Functions and Vertical AsymptotesRational Functions and HolesEquivalent Representations of Polynomial and Rational ExpressionsTransformations of FunctionsFunction Model Selection and Assumption ArticulationFunction Model Construction and ApplicationChange in Arithmetic and Geometric SequencesChange in Linear and Exponential FunctionsExponential FunctionsExponential Function ManipulationExponential Function Context and Data ModelingCompeting Function Model ValidationComposition of FunctionsInverse FunctionsLogarithmic ExpressionsInverses of Exponential FunctionsLogarithmic FunctionsLogarithmic Function ManipulationExponential and Logarithmic Equations and InequalitiesLogarithmic Function Context and Data ModelingSemi-log PlotsPeriodic PhenomenaSine, Cosine, and TangentSine and Cosine Function ValuesSine and Cosine Function GraphsSinusoidal FunctionsSinusoidal Function TransformationsSinusoidal Function Context and Data ModelingThe Tangent FunctionInverse Trigonometric FunctionsTrigonometric Equations and InequalitiesThe Secant, Cosecant, and Cotangent FunctionsEquivalent Representations of Trigonometric FunctionsTrigonometry and Polar CoordinatesPolar Function GraphsRates of Change in Polar FunctionsParametric FunctionsParametric Functions Modeling Planar MotionParametric Functions and Rates of ChangeParametrically Defined Circles and LinesImplicitly Defined FunctionsConic SectionsParametrization of Implicitly Defined FunctionsVectorsVector-Valued FunctionsMatricesThe Inverse and Determinant of a MatrixLinear Transformations and MatricesMatrices as FunctionsMatrices Modeling Contexts
Quizzes
Practice Problems Open a unit to choose from 58 topic-aligned practice sets.

Unit 6 · Topic 3.13

Trigonometry and Polar Coordinates

Describe the same point by rectangular displacement or by directed distance and angle, then connect both descriptions to complex numbers.

Learning Goals

  • Interpret the polar coordinates \((r,\theta)\) on a polar grid.
  • Explain positive and negative angles and signed radii.
  • Generate multiple polar representations of the same point.
  • Convert points and simple equations between polar and rectangular forms.
  • Write complex numbers in rectangular and polar form.

1. Two Coordinate Systems, One Plane

Rectangular coordinates \((x,y)\) record horizontal and vertical displacement from the origin. Polar coordinates \((r,\theta)\) record a directed radial displacement and an angle measured from the positive x-axis.

Rectangular languagePolar language
OriginPole
Positive x-axisPolar axis
Horizontal and vertical displacementRadius and direction

2. Read the Ordered Pair Correctly

In \((r,\theta)\), the radius is written first even though it is helpful to choose the direction \(\theta\) before measuring the directed distance \(r\).

  1. Begin on the polar axis.
  2. Rotate counterclockwise for a positive angle or clockwise for a negative angle.
  3. Move \(|r|\) units along or opposite that direction according to the sign of \(r\).

3. Understand a Negative Radius

If \(r>0\), move in the direction of \(\theta\). If \(r<0\), move \(|r|\) units in the opposite direction.

\[(-r,\theta)\quad\text{and}\quad(r,\theta+\pi)\]

represent the same point when \(r>0\). A negative radius changes the final direction by \(\pi\); it does not mean that the point has a negative distance from the pole.

4. One Point Has Infinitely Many Polar Names

For any integer \(k\),

\[(r,\theta)\equiv(r,\theta+2k\pi)\]
\[(r,\theta)\equiv(-r,\theta+(2k+1)\pi).\]

For example, \((3,\pi/6)\), \((3,13\pi/6)\), and \((-3,7\pi/6)\) all locate the same point.

5. Connect Radius, Angle, and Components

Trigonometry and Polar Coordinates example graphThe angle sets direction and the radius sets directed distance from the pole.
The angle sets direction and the radius sets directed distance from the pole.

The radial segment and its horizontal and vertical components form a right triangle. Cosine gives the horizontal fraction of \(r\), and sine gives the vertical fraction.

6. Convert Polar to Rectangular

From \(\cos\theta=x/r\) and \(\sin\theta=y/r\),

\[x=r\cos\theta,\qquad y=r\sin\theta.\]

These formulas work for positive or negative \(r\). The signs of the products automatically place the point in the correct quadrant.

7. Polar-to-Rectangular Example

Convert \((r,\theta)=(4,5\pi/6)\):

\[x=4\cos\left(\frac{5\pi}{6}\right)=4\left(-\frac{\sqrt3}{2}\right)=-2\sqrt3,\]
\[y=4\sin\left(\frac{5\pi}{6}\right)=4\left(\frac12\right)=2.\]

Thus the rectangular coordinates are \((-2\sqrt3,2)\), consistent with Quadrant II.

8. Convert a Point with Negative Radius

Convert \((-3,\pi/4)\):

\[x=-3\cos\left(\frac{\pi}{4}\right)=-\frac{3\sqrt2}{2},\qquad y=-3\sin\left(\frac{\pi}{4}\right)=-\frac{3\sqrt2}{2}.\]

The final point is in Quadrant III, opposite the \(\pi/4\) ray. An equivalent positive-radius representation is \((3,5\pi/4)\).

9. Convert Rectangular to Polar

Use the Pythagorean relationship and trigonometric ratios:

\[r=\sqrt{x^2+y^2},\qquad \cos\theta=\frac{x}{r},\qquad\sin\theta=\frac{y}{r}.\]
\[\tan\theta=\frac{y}{x}\qquad(x\ne0).\]

The square root gives the conventional choice \(r\ge0\). Determine the point's quadrant before selecting \(\theta\).

10. Correct the Arctangent Quadrant

The calculator value \(\arctan(y/x)\) lies only in \((-\pi/2,\pi/2)\), so the coordinate signs must guide the final angle.

Coordinate conditionAngle strategy
\(x>0\)Use \(\arctan(y/x)\), adjusted to the desired angle interval
\(x<0\)Add \(\pi\) to place the angle in Quadrant II or III
\(x=0,\ y>0\)\(\theta=\pi/2\)
\(x=0,\ y<0\)\(\theta=3\pi/2\) or \(-\pi/2\)

11. Rectangular-to-Polar Example

Convert \((-3,3\sqrt3)\) using \(r\ge0\) and \(0\le\theta<2\pi\):

\[r=\sqrt{(-3)^2+(3\sqrt3)^2}=\sqrt{9+27}=6.\]

The point is in Quadrant II and has reference angle \(\pi/3\), so

\[\theta=\frac{2\pi}{3},\qquad(r,\theta)=\left(6,\frac{2\pi}{3}\right).\]

Substitution gives \(6\cos(2\pi/3)=-3\) and \(6\sin(2\pi/3)=3\sqrt3\).

12. Handle the Pole Separately

At the rectangular origin, \(r=0\). The angle has no unique direction, so

\[(0,\theta)\]

represents the pole for every real \(\theta\). The ratio \(y/x\) cannot determine an angle there.

13. Convert a Simple Polar Equation

Convert \(r=4\cos\theta\) to rectangular form. Multiply by \(r\):

\[r^2=4r\cos\theta.\]

Use \(r^2=x^2+y^2\) and \(r\cos\theta=x\):

\[x^2+y^2=4x\quad\Longleftrightarrow\quad(x-2)^2+y^2=4.\]

The polar equation represents a circle centered at \((2,0)\) with radius 2.

14. Connect Polar Coordinates to Complex Numbers

The complex number \(z=a+bi\) corresponds to the point \((a,b)\). If that point has polar coordinates \((r,\theta)\), then

\[z=r\cos\theta+i(r\sin\theta)=r(\cos\theta+i\sin\theta).\]

For \(z=-2+2\sqrt3\,i\),

\[r=\sqrt{(-2)^2+(2\sqrt3)^2}=4,\qquad\theta=\frac{2\pi}{3},\]
\[z=4\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right).\]

Here \(r=|z|\) is the modulus and \(\theta\) is an argument of the complex number.

15. AP Workflow and Common Errors

  1. Identify the starting coordinate system.
  2. Sketch the quadrant or directed ray.
  3. Use \(x=r\cos\theta\) and \(y=r\sin\theta\) for polar-to-rectangular conversion.
  4. Use \(r=\sqrt{x^2+y^2}\) and coordinate signs for rectangular-to-polar conversion.
  5. Verify by converting back.
  6. State the requested interval for \(\theta\) and sign convention for \(r\).
  • Do not swap \(r\) and \(\theta\) in the ordered pair.
  • Do not treat a negative radius as an ordinary negative distance.
  • Do not trust \(\arctan(y/x)\) without checking the quadrant.
  • Do not claim a polar representation is unique.
  • Do not use \(r=x^2+y^2\) instead of \(r^2=x^2+y^2\).

Key Takeaways

  • Convert points and equations between rectangular and polar coordinate systems.
  • Core relationship: \(x=r\cos\theta,\quad y=r\sin\theta,\quad r^2=x^2+y^2\)
  • Error check: Remember that negative radii describe points in the opposite direction.
Checkpoint · Topic 3.13
  1. Plot \((-2,\pi/3)\) and write an equivalent representation with positive radius.
  2. Convert \((6,7\pi/6)\) to rectangular coordinates.
  3. Convert \((4,-4)\) to polar coordinates with \(r\ge0\) and \(0\le\theta<2\pi\).
  4. Convert \(r=6\sin\theta\) to rectangular form and identify the curve.
  5. Write \(z=-3-3i\) in polar form using an argument in \([0,2\pi)\).