Welcome to the World of Waves!
In this chapter, we explore circular functions—more commonly known as trigonometric functions like sine, cosine, and tangent. Why "circular"? Because they are based on the coordinates of a point moving around the unit circle. These functions are the mathematical heartbeat of anything that repeats, from the rise and fall of ocean tides to the sound waves coming out of your headphones. Don't worry if these curvy graphs look intimidating at first; we will break them down into simple parts that you can master.
1. The Parent Graphs: Sine, Cosine, and Tangent
Before we start moving graphs around, we need to know what the "base" or "parent" versions look like. We usually measure the x-axis in radians (like \(\pi, 2\pi\)) rather than degrees.
The Sine Function: \(y = \sin x\)
Think of the sine graph as a wave that starts at the origin \((0,0)\).
- Shape: It starts at \(0\), goes up to a maximum of \(1\), back to \(0\), down to a minimum of \(-1\), and returns to \(0\).
- Period: The wave repeats every \(2\pi\) units.
- Amplitude: The height from the middle to the top is \(1\).
- Domain: \(x \in \mathbb{R}\) (all real numbers).
- Range: \(-1 \leq y \leq 1\).
The Cosine Function: \(y = \cos x\)
The cosine graph looks almost exactly like sine, but it is "shifted."
- Shape: It starts at its maximum value \((0,1)\), goes down to \(-1\), and comes back up.
- Period: \(2\pi\).
- Amplitude: \(1\).
- Range: \(-1 \leq y \leq 1\).
Tip: If sine is a "mountain-valley" starting at zero, cosine is a "cup" shape starting at the peak.
The Tangent Function: \(y = \tan x\)
Tangent is the "rebel" of the group. It doesn't look like a wave; it looks like a series of repeating curves separated by vertical lines called asymptotes.
- Period: \(\pi\) (Note: this is half the period of sine and cosine!).
- Asymptotes: These occur where \(\cos x = 0\), such as at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\). The graph never touches these lines.
- Range: \(y \in \mathbb{R}\) (It goes from \(-\infty\) to \(+\infty\)).
Quick Review: Sine and Cosine repeat every \(2\pi\). Tangent repeats every \(\pi\).
2. Transforming the Waves: \(y = a \sin(b(x + c)) + d\)
In the IB exam, you will often see sine and cosine functions that have been stretched or shifted. Each letter in the formula \(y = a \sin(b(x + c)) + d\) (or the cosine version) changes the graph in a specific way.
\(a\) : The Amplitude (Vertical Stretch)
The value of \(|a|\) tells you the amplitude.
- If \(a = 3\), the wave goes up to \(3\) and down to \(-3\).
- If \(a\) is negative, the graph is reflected across the x-axis.
\(d\) : The Midline / Principal Axis (Vertical Translation)
The value of \(d\) shifts the entire graph up or down.
- It is the horizontal line that runs exactly through the middle of the wave.
- Formula Tip: \(d = \frac{\text{max} + \text{min}}{2}\).
\(b\) : The Frequency Factor (Horizontal Stretch)
This is the most common place for mistakes! The value of \(b\) is not the period, but it helps you calculate it.
- Period formula: \(\text{Period} = \frac{2\pi}{b}\) (for sine and cosine).
- Period formula for tangent: \(\text{Period} = \frac{\pi}{b}\).
- If \(b\) is large, the waves are "squished" together. If \(b\) is small (like \(0.5\)), the waves are stretched out.
\(c\) : The Phase Shift (Horizontal Translation)
The value of \(c\) moves the graph left or right.
- In the form \((x + c)\), the graph moves Left if \(c > 0\).
- In the form \((x + c)\), the graph moves Right if \(c < 0\).
Key Takeaway: Always factor out \(b\) to see the phase shift clearly. For example, in \(\sin(2x + \pi)\), rewrite it as \(\sin(2(x + \frac{\pi}{2}))\). The shift is \(\frac{\pi}{2}\) to the left.
3. Real-Life Contexts
Circular functions are perfect for modeling "periodic" behavior. You might be asked to find a function for:
- Ferris Wheels: The height of a rider over time.
- Tides: The depth of water in a harbor over 24 hours.
- Temperatures: The daily fluctuation of heat in a city.
Step-by-Step for Modeling:
1. Find the Midline (\(d\)): Halfway between the max and min.
2. Find the Amplitude (\(a\)): Distance from midline to max.
3. Find the Period: Time taken for one full cycle. Use it to find \(b = \frac{2\pi}{\text{Period}}\).
4. Find the Shift (\(c\)): Look at where the cycle starts compared to the parent graph.
4. Higher Level (AHL) Only: Reciprocal and Inverse Graphs
If you are an HL student, you need to be familiar with a few extra curves.
Inverse Trigonometric Graphs
Because trig functions repeat, they don't have inverses unless we restrict their domains.
- \(y = \arcsin x\) (or \(\sin^{-1} x\)): Domain is \([-1, 1]\), Range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
- \(y = \arccos x\) (or \(\cos^{-1} x\)): Domain is \([-1, 1]\), Range is \([0, \pi]\).
- \(y = \arctan x\) (or \(\tan^{-1} x\)): Domain is \(\mathbb{R}\), Range is \((-\frac{\pi}{2}, \frac{\pi}{2})\). This graph has horizontal asymptotes at \(y = \pm \frac{\pi}{2}\).
Symmetry Properties (AHL)
Graphs also show us the "even" and "odd" nature of these functions:
- Sine is Odd: \(\sin(-x) = -\sin x\). The graph has rotational symmetry about the origin.
- Cosine is Even: \(\cos(-x) = \cos x\). The graph is symmetrical across the y-axis.
5. Common Pitfalls to Avoid
- Degrees vs Radians: Most graphing questions are in radians. Ensure your calculator is in the correct mode!
- Amplitude is always positive: While the value of \(a\) in an equation can be negative (indicating a reflection), the amplitude itself is defined as the absolute distance \(|a|\).
- Tangent Period: Remember that \(\tan x\) has a period of \(\pi\), not \(2\pi\). This is a very common mistake in Paper 1 and Paper 2.
Did you know? The word "sine" comes from the Latin word sinus, meaning "bay" or "curve." It perfectly describes the smooth, flowing shape of the graph!
Summary Checklist
- Can you sketch \(\sin x, \cos x,\) and \(\tan x\) from memory?
- Can you identify the amplitude, period, and midline from an equation?
- For HL: Do you know the restricted ranges for the inverse functions?
- Can you use a GDC to find the points of intersection between two trig graphs?