Graphs of Sine and Cosine Revision Notes for VCE SSCE Mathematical Methods

Graphs of Sine and Cosine

Introduction

The sine and cosine functions are fundamental periodic functions that repeat their patterns at regular intervals. Understanding their graphs and how to transform them is essential for working with circular functions.

Graph of the sine function

The basic sine function $f(x) = \sin x$ produces a smooth, wave-like curve. Here's what it looks like:

Key properties of $y = \sin x$

Periodic nature

When a function repeats the same pattern over and over, we call it a periodic function. The distance along the x-axis before the pattern repeats is called the period of the function. Some textbooks also refer to this as the wavelength.

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For $y = \sin x$ , the period is $2\pi$ units. This means the graph completes one full cycle every $2\pi$ units along the x-axis.

Amplitude

The amplitude is the distance between the middle position (the x-axis) and the maximum (or minimum) position of the wave.

For $y = \sin x$ :

Maximum value = $1$
Minimum value = $-1$
Amplitude = 1

Graph of the cosine function

The basic cosine function $g(x) = \cos x$ also produces a wave-like curve:

Key properties of $y = \cos x$

Period and amplitude

Like the sine function:

Period = $2\pi$
Amplitude = $1$

Relationship to sine

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The graph of $y = \cos x$ is identical to the graph of $y = \sin x$ shifted horizontally. Specifically, if you move the sine graph π/2 units to the left (in the negative x-direction), you get the cosine graph.

Transformations: dilations

We can transform the basic sine and cosine graphs by changing their amplitude and period. This is done using functions of the form:

y = a\sin(nt) \quad \text{and} \quad y = a\cos(nt)

where $a$ and $n$ are positive numbers.

Understanding the transformations

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Effect of the coefficient $a$

The coefficient $a$ controls the amplitude:

It stretches or compresses the graph vertically
The new amplitude equals $a$
Larger values of $a$ make taller waves
The graph oscillates between $-a$ and $a$

Effect of the coefficient $n$

The coefficient $n$ controls the period:

It compresses or stretches the graph horizontally
The new period equals $\frac{2\pi}{n}$
Larger values of $n$ create more cycles in the same space
The graph completes one full cycle over a distance of $\frac{2\pi}{n}$

Worked example: $y = 3\sin(2t)$

Let's see how these transformations work with a specific example.

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Worked Example: Analyzing the function $y = 3\sin(2t)$

For this function:

Amplitude = 3 (the wave reaches up to $3$ and down to $-3$ )
Period = π (calculated as $\frac{2\pi}{2} = \pi$ , one complete cycle occurs over a distance of $\pi$ )

The graph is obtained from $y = \sin t$ by:

A vertical dilation (stretch) by a factor of $3$ from the t-axis
A horizontal dilation (compression) by a factor of $\frac{1}{2}$ from the y-axis

Worked example: $y = 2\cos(3t)$

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Worked Example: Analyzing the function $y = 2\cos(3t)$

For this function:

Amplitude = 2
Period = 2π/3 (calculated as $\frac{2\pi}{3}$ )

The graph is obtained from $y = \cos t$ by:

A vertical dilation by a factor of $2$ from the t-axis
A horizontal dilation by a factor of $\frac{1}{3}$ from the y-axis

Amplitude and period formulas

Here's a summary table of the key formulas:

For any function of the form $y = a\sin(nt)$ or $y = a\cos(nt)$ where $a$ and $n$ are positive:

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Key Formulas:

Amplitude = $a$

Period = $\frac{2\pi}{n}$

Domain = $\mathbb{R}$ (all real numbers)

Range = $[-a, a]$

Worked example: finding amplitude and period

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Worked Example: Finding amplitude and period

Question: For each of the following functions with domain $\mathbb{R}$ , state the amplitude and period:

a) $f(t) = 2\sin(3t)$

Amplitude = 2

Period = 2π/3 (calculated as $\frac{2\pi}{3}$ )

b) $f(t) = -\frac{1}{2}\sin\left(\frac{t}{2}\right)$

Amplitude = 1/2 (take the absolute value of the coefficient)

Period = 4π (calculated as $2\pi \div \frac{1}{2} = 4\pi$ )

c) $f(t) = 4\cos(3\pi t)$

Amplitude = 4

Period = 2/3 (calculated as $\frac{2\pi}{3\pi} = \frac{2}{3}$ )

Sketching transformed graphs

When sketching graphs of transformed sine and cosine functions, follow these steps:

Identify the amplitude (how high and low the wave goes)
Calculate the period (how long one complete cycle is)
Mark key points on the horizontal axis
Sketch one complete cycle
Label the function and axes

Worked example: sketching sine and cosine graphs

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Worked Example: Sketching graphs

Question: Sketch the graph of each of the following functions, showing one complete cycle:

a) $y = 2\cos(2\theta)$

The amplitude is 2.

The period is π (calculated as $\frac{2\pi}{2} = \pi$ ).

The graph starts at its maximum value of 2 when $\theta = 0$ , decreases to $-2$ at $\theta = \frac{\pi}{2}$ , and returns to $2$ at $\theta = \pi$ .

b) $y = \frac{1}{2}\sin\left(\frac{x}{2}\right)$

The amplitude is 1/2.

The period is 4π (calculated as $2\pi \div \frac{1}{2} = 4\pi$ ).

The graph starts at $0$ when $x = 0$ , reaches its maximum of 1/2 at $x = \pi$ , returns to $0$ at $x = 2\pi$ , reaches its minimum of $-\frac{1}{2}$ at $x = 3\pi$ , and completes the cycle at $x = 4\pi$ .

Transformations: reflections

When the coefficient $a$ is negative, the graph is reflected in the x-axis.

Properties of negative coefficients

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For $y = -a\sin(nt)$ or $y = -a\cos(nt)$ :

The amplitude is still $a$ (take the absolute value)
The period is still $\frac{2\pi}{n}$
The graph is flipped upside down compared to the positive version

Worked example: graphs with reflections

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Worked Example: Graphing with reflections

Question: Sketch the following graphs for $x \in [0, 4\pi]$ :

a) $f(x) = -2\sin\left(\frac{x}{2}\right)$

The graph is obtained from $y = 2\sin\left(\frac{x}{2}\right)$ by reflecting in the x-axis.

Amplitude = 2

Period = 4π

b) $y = -\cos(2x)$

The graph is obtained from $y = \cos(2x)$ by reflecting in the x-axis.

Amplitude = 1

Period = π

Important note about odd and even functions

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Recall that:

Sine is an odd function: $\sin(-x) = -\sin x$
Cosine is an even function: $\cos(-x) = \cos x$

This means:

When you reflect $y = \sin x$ in the y-axis, you get $y = -\sin x$
When you reflect $y = \cos x$ in the y-axis, the graph stays the same

Summary of properties

For positive numbers $a$ and $n$ , the functions $y = a\sin(nt)$ , $y = -a\sin(nt)$ , $y = a\cos(nt)$ , and $y = -a\cos(nt)$ all share these properties:

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Key Properties to Remember:

Period = $\frac{2\pi}{n}$

Amplitude = $a$

Domain = $\mathbb{R}$ (all real numbers)

Range = $[-a, a]$

Remember!

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Essential Takeaways:

Sine and cosine are periodic functions that repeat every $2\pi$ units in their basic form
The amplitude tells you how high and low the wave goes from the centre line
The period tells you how long it takes for one complete cycle
For $y = a\sin(nt)$ or $y = a\cos(nt)$ : amplitude = a and period = 2π/n
A negative coefficient reflects the graph in the x-axis but doesn't change the amplitude or period
The cosine graph is just the sine graph shifted π/2 units to the left

Graphs of Sine and Cosine (VCE SSCE Mathematical Methods): Revision Notes