AP Precalculus Section 3.15 Example: Plot a Complex Polar Coordinate on a Complex Plane

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Plotting a complex polar coordinate on a complex plane involves expressing a complex number in its polar form \((r, \theta)\) and then representing it on the plane. The polar form of a complex number \(z\) is given by:

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

Here's how you can plot a complex polar coordinate on a complex plane:

1. **Identify \(r\) and \(\theta\):**
- Given a complex number in polar form \(z = r(\cos \theta + i \sin \theta)\), identify the magnitude \(r\) and the angle \(\theta\) with the positive x-axis.

2. **Express in Rectangular Form:**
- Use Euler's formula to express the polar form in rectangular form: \(z = r \cos \theta + i r \sin \theta\).

3. **Plot on Complex Plane:**
- Represent the real part \(x = r \cos \theta\) along the x-axis and the imaginary part \(y = r \sin \theta\) along the y-axis.
- Place a point at the coordinates \((x, y)\) on the complex plane.

4. **Include Magnitude and Angle:**
- Optionally, draw a line from the origin to the plotted point, representing the magnitude \(r\).
- Indicate the angle \(\theta\) with the positive x-axis.

For example, if you have the complex number \(z = 3(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\):
- Express in rectangular form: \(z = 3 \cos \frac{\pi}{4} + i \cdot 3 \sin \frac{\pi}{4}\).
- Simplify: \(z = \frac{3}{\sqrt{2}} + i \cdot \frac{3}{\sqrt{2}}\).
- Plot the point \(\left(\frac{3}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)\) on the complex plane.

Remember that the polar form provides a way to express a complex number in terms of its magnitude (\(r\)) and angle (\(\theta\)). The rectangular form allows for easy visualization on the complex plane.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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