Welcome to the World of Cycles!
Welcome to Unit 3! In this unit, we are moving beyond simple triangles and entering the world of periodic functions. These are functions that repeat their values at regular intervals. Why does this matter? Because almost everything in nature happens in cycles—the tides, your heartbeat, sound waves, and even the phases of the moon. By the end of these notes, you'll be able to describe these patterns using Trigonometric and Polar Functions. Don't worry if it seems like a lot of math symbols at first; we will break it down step-by-step!
3.1 & 3.2: Radians, Circles, and The Big Three
In geometry, you used degrees. In AP Precalculus, we prefer radians. Think of radians as measuring the distance you actually travel along the edge of a circle.
Key Concept: The Radian
A full circle is \(360^\circ\), which is equal to \(2\pi\) radians.
• To go from degrees to radians: Multiply by \(\frac{\pi}{180}\).
• To go from radians to degrees: Multiply by \(\frac{180}{\pi}\).
The Unit Circle: Your Best Friend
The Unit Circle is a circle with a radius of 1. Any point on this circle has coordinates \((x, y)\), where:
• \(x = \cos(\theta)\) (The horizontal distance)
• \(y = \sin(\theta)\) (The vertical distance)
• \(\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\) (The slope of the line)
Analogy: Imagine you are on a Ferris wheel. The Sine is how high you are above the ground, and the Cosine is how far you are left or right from the center pole.
Quick Review: Signs of Functions
Use the mnemonic "All Students Take Calculus" to remember which functions are positive in each quadrant:
• Quadrant I: All (Sin, Cos, Tan) are positive.
• Quadrant II: Sine is positive.
• Quadrant III: Tangent is positive.
• Quadrant IV: Cosine is positive.
Key Takeaway: Every angle on the unit circle gives us a specific \((x, y)\) coordinate, which defines our Sine and Cosine values.
3.3 - 3.5: Graphing Sine and Cosine
When we "unroll" the unit circle onto a flat graph, we get a Sinusoidal Wave. These waves look like smooth hills and valleys.
The Anatomy of a Wave
For the function \(f(x) = a \sin(b(x - c)) + d\):
1. Amplitude (\(|a|\)): This is the "height" of the wave from the center. It’s half the distance between the max and min.
2. Midline (\(y = d\)): The horizontal line that runs through the exact middle of the graph.
3. Period: The horizontal distance it takes to complete one full cycle. Formula: \(P = \frac{2\pi}{|b|}\).
4. Phase Shift (\(c\)): The horizontal "slide" left or right.
5. Frequency: How many cycles happen in a standard interval (\(2\pi\)). This is represented by \(b\).
Step-by-Step: How to Graph
1. Draw your midline (\(d\)).
2. Go up and down from the midline by the amplitude (\(a\)) to find your max and min heights.
3. Calculate the period (\(\frac{2\pi}{b}\)) to see how wide one wave is.
4. Divide the period into 4 equal chunks to find your "key points" (Intercept, Max, Intercept, Min, Intercept).
Common Mistake: Many students forget that the period is not \(b\). You must divide \(2\pi\) by \(b\) to find the period!
Key Takeaway: Sine and Cosine are the same shape; Cosine is just a Sine wave shifted to the left by \(\frac{\pi}{2}\).
3.6: Modeling with Trig Functions
This is where math meets the real world! If you have data that goes up and down regularly (like temperature over a year), you can create an equation for it.
Example: A Ferris wheel starts at its lowest point (2 meters off the ground), reaches a max height of 22 meters, and takes 60 seconds for a full rotation.
• Midline: \(\frac{22 + 2}{2} = 12\). So, \(d = 12\).
• Amplitude: \(22 - 12 = 10\). So, \(a = 10\).
• Period: 60 seconds. Since \(P = \frac{2\pi}{b}\), then \(60 = \frac{2\pi}{b}\), so \(b = \frac{\pi}{30}\).
3.7: The Tangent Function
The graph of \(\tan(x)\) looks very different. It doesn't have "hills and valleys"—it has vertical asymptotes.
Why? Remember \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Whenever \(\cos(x) = 0\) (at \(\frac{\pi}{2}, \frac{3\pi}{2}\), etc.), the function is dividing by zero. This creates a "wall" or asymptote.
Key Features of Tangent:
• Period: \(\frac{\pi}{|b|}\) (Notice it’s \(\pi\), not \(2\pi\)!)
• Range: All real numbers (\((-\infty, \infty)\)).
• Domain: All real numbers except where \(\cos(x) = 0\).
3.8: Inverse Trig Functions
Normally, we put an angle into a trig function and get a ratio (like \(\sin(30^\circ) = 0.5\)).
Inverse functions do the opposite: you give them the ratio, and they give you the angle (\(\arcsin(0.5) = 30^\circ\)).
The Catch: Because trig functions repeat forever, they fail the "Horizontal Line Test." To fix this, we restrict the answers (ranges):
• \(\arcsin(x)\) and \(\arctan(x)\): Only give answers between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) (Right side of the circle).
• \(\arccos(x)\): Only gives answers between \(0\) and \(\pi\) (Top half of the circle).
Don't worry if this seems tricky! Just remember: The inverse function is asking "What angle gives me this value?" but it’s only allowed to look in a small "parking lot" of angles.
3.10: Trigonometric Identities
Identities are equations that are always true. They are the "tools" you use to simplify complex expressions.
The "Must-Know" Identities:
1. Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
2. Reciprocal Identities:
• \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
Did you know? The Pythagorean identity is just the Pythagorean theorem (\(a^2 + b^2 = c^2\)) applied to the unit circle where the radius (hypotenuse) is 1!
3.11 - 3.12: Polar Coordinates and Functions
Up until now, you've lived in a Rectangular world \((x, y)\). Now, welcome to the Polar world!
In Polar coordinates, we identify a point by \((r, \theta)\):
• \(r\): The distance from the center (origin).
• \(\theta\): The angle from the positive x-axis.
Converting between the two:
• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)
• \(r^2 = x^2 + y^2\)
• \(\tan(\theta) = \frac{y}{x}\)
Polar Graphs
Polar graphs create beautiful shapes like circles, roses, and limaçons.
• If \(r = 3\), it's just a circle with radius 3.
• If \(\theta = \frac{\pi}{4}\), it's a straight line at that angle.
• Equations like \(r = a \sin(n\theta)\) create "Rose curves" where \(n\) tells you how many petals the flower has.
Key Takeaway: Polar coordinates are great for things that rotate or radiate out from a center, like radar or flower petals.
3.13: Rates of Change
Just like with linear functions, we can look at how fast trig or polar functions are changing.
• If the graph is steep, the Rate of Change is high.
• If the graph is flat (like at the top of a Sine wave), the Rate of Change is zero.
• In polar functions, we often look at how the radius \(r\) changes as the angle \(\theta\) increases. If \(\frac{\Delta r}{\Delta \theta}\) is positive, the point is moving further away from the center.
Final Encouragement: Trigonometry is a language of patterns. Once you learn the "alphabet" (the unit circle) and the "grammar" (identities and transformations), you'll be able to describe the rhythmic movements of the entire universe. Keep practicing your unit circle—it's the key to everything!