Graphs and Equations (HSC SSCE Mathematics Advanced): Revision Notes

Trigonometric Graphs

Introduction to radian measure

When working with trigonometric functions in mathematics, we measure angles in radians rather than degrees. A radian is a pure number without units, making it ideal for calculus and advanced mathematics.

The key conversions between radians and degrees are:

$2\pi = 360°, \quad \pi = 180°, \quad \frac{\pi}{2} = 90°, \quad \frac{\pi}{3} = 60°, \quad \frac{\pi}{4} = 45°, \quad \frac{\pi}{6} = 30°$

The three fundamental trigonometric graphs

There are three primary trigonometric functions whose graphs you need to understand: sine, cosine and tangent. Each has distinct characteristics that make it unique.

The sine function

The graph of $y = \sin x$ is a smooth wave that oscillates between 1 and -1. It crosses the $x$-axis at multiples of $\pi$ (that is, at $0, \pm\pi, \pm2\pi, \pm3\pi$, and so on).

The cosine function

The graph of $y = \cos x$ is also a smooth wave oscillating between $1$ and $-1$, but it starts at its maximum value when $x = 0$. The cosine graph looks like the sine graph shifted horizontally.

The tangent function

The graph of $y = \tan x$ is quite different. Instead of being bounded like sine and cosine, the tangent function has vertical asymptotes (lines the graph approaches but never touches) at $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}$, and so on.

Between each pair of asymptotes, the tangent curve increases from negative infinity to positive infinity, creating a repeating pattern of S-shaped curves.

Vertical dilations and amplitude

The amplitude of a wave measures how far above the middle position the wave reaches at its peak. For the basic functions $y = \sin x$ and $y = \cos x$, the waves oscillate between $1$ and $-1$, with a middle position at $y = 0$. This means both have amplitude $1$.

When we multiply a sine or cosine function by a constant $a$, we stretch the graph vertically. The function $y = a\sin x$ will now oscillate between $a$ and $-a$, giving it an amplitude of $a$.

Key properties:

• $y = \sin x$ and $y = \cos x$ both have amplitude $1$
• $y = a\sin x$ and $y = a\cos x$ both have amplitude $a$
• These are created by stretching $y = \sin x$ or $y = \cos x$ vertically by factor $a$

The tangent function increases without bound near its asymptotes, so amplitude doesn't apply. However, we can still stretch it vertically to get $y = a\tan x$. For this function, when $x = \frac{\pi}{4}$, we have $y = a$.

Horizontal dilations and period

Periodic functions are functions that repeat themselves over and over. The period is the length of the smallest repeating section of the graph.

For the basic trigonometric functions:

• $y = \sin x$ and $y = \cos x$ complete one full wave cycle every $2\pi$ units (one full revolution), so they have period $2\pi$
• $y = \tan x$ repeats every $\pi$ units (half a revolution), so it has period $\pi$

When we introduce a coefficient inside the function, we compress the graph horizontally. Consider $y = \sin nx$. This can be written as $y = \sin\frac{x}{1/n}$, showing it's a horizontal compression by factor $\frac{1}{n}$.

Since the original function $y = \sin x$ has period $2\pi$, the compressed function $y = \sin nx$ has period $\frac{2\pi}{n}$.

Horizontal translations and phase

The phase (or initial phase angle) of a trigonometric function tells us the angle value when $x = 0$. For example, $y = \sin(x + \frac{\pi}{3})$ has phase $\frac{\pi}{3}$ because when $x = 0$, the angle inside the sine function is $0 + \frac{\pi}{3} = \frac{\pi}{3}$.

When we add a constant $\alpha$ inside the function, we shift the graph horizontally. The function $y = \sin(x + \alpha)$ is the result of shifting $y = \sin x$ to the left by $\alpha$ units.

Key properties:

• The phase is the angle when $x = 0$
• $y = \sin(x + \alpha)$, $y = \cos(x + \alpha)$ and $y = \tan(x + \alpha)$ all have phase $\alpha$
• These functions are created by shifting left by $\alpha$ (if $\alpha > 0$) or right by $|\alpha|$ (if $\alpha < 0$)

Combining period and phase

When we have both a horizontal compression and a horizontal shift, we need to be careful about the order of transformations. Consider the function:

$y = \sin(2x + \frac{\pi}{3}) \quad \text{or equivalently} \quad y = \sin 2(x + \frac{\pi}{6})$

This function has:

• Period: $\frac{2\pi}{2} = \pi$
• Phase: $\frac{\pi}{3}$ (found by setting $x = 0$)

There are two ways to think about creating this graph:

Method 1: Starting from $y = \sin(2x + \frac{\pi}{3})$

• Shift $y = \sin x$ left by $\frac{\pi}{3}$ to get $y = \sin(x + \frac{\pi}{3})$
• Then compress horizontally by factor $\frac{1}{2}$ to get $y = \sin(2x + \frac{\pi}{3})$

Method 2: Starting from $y = \sin 2(x + \frac{\pi}{6})$

• Compress $y = \sin x$ horizontally by factor $\frac{1}{2}$ to get $y = \sin 2x$
• Then shift left by $\frac{\pi}{6}$ to get $y = \sin 2(x + \frac{\pi}{6})$

General formula: For $y = \sin(nx + \alpha)$:

• Period: $\frac{2\pi}{n}$
• Phase: $\alpha$

Vertical translations and mean value

The mean value of a wave is the average of its maximum and minimum values. For $y = \sin x$ and $y = \cos x$, the maximum is $1$ and the minimum is $-1$, so the mean value is $\frac{1 + (-1)}{2} = 0$.

When we add a constant $c$ to a trigonometric function, we shift it vertically. The function $y = \sin x + c$ has:

• Mean value: $c$
• The wave still has the same amplitude, but it now oscillates around $y = c$ instead of $y = 0$

Key properties:

• $y = \sin x + c$ and $y = \cos x + c$ both have mean value $c$
• These are created by shifting up by $c$ units (or down if $c \< 0$)
• When combining transformations, do vertical translations last to avoid confusion with amplitude changes

The function $y = \tan x + c$ isn't a wave and doesn't have a mean value, but it's still shifted up by $c$ units.

Symmetry properties of trigonometric functions

The three basic trigonometric functions have important symmetry properties that affect how they behave with negative inputs.

Odd functions: sine and tangent

The functions $y = \sin x$ and $y = \tan x$ are odd functions. This means:

$\sin(-x) = -\sin x \quad \text{and} \quad \tan(-x) = -\tan x$

Graphically, odd functions have point symmetry about the origin. If you rotate the graph $180°$ about the origin, it looks unchanged.

Even function: cosine

The function $y = \cos x$ is an even function. This means:

$\cos(-x) = \cos x$

Graphically, even functions have line symmetry in the y-axis. If you reflect the graph across the $y$-axis, it looks unchanged.

These symmetry properties are extremely useful when solving equations and simplifying expressions.

Graphical solutions of trigonometric equations

Many trigonometric equations cannot be solved using algebra alone. However, we can use graphs to find approximate solutions. By sketching both sides of an equation on the same axes, we can:

• Determine how many solutions exist
• Find approximate values for the solutions