7. Trigonometry

iGCSE (2021 Edition)

Lesson

The graphs of trigonometric functions $y=\sin x$`y`=`s``i``n``x`, $y=\cos x$`y`=`c``o``s``x`, and $y=\tan x$`y`=`t``a``n``x` have repeated sections. Functions that repeat themselves in fixed intervals are called periodic. The graphs of $y=\sin x$`y`=`s``i``n``x` and $y=\cos x$`y`=`c``o``s``x` are defined by their amplitude, phase and period, while the graph of $y=\tan x$`y`=`t``a``n``x` is defined by its period and phase only, as it does not have maximum or minimum values.

In this lesson, we will define amplitude and consider the transformation of trigonometric graphs through vertical shifts and dilations. In subsequent lessons we will look at the definition of the **period** and **phase shifts** of trigonometric functions.

The sine and cosine functions have a minimum value, a maximum value, and an average value about which they oscillate. Both of the basic functions $y=\sin x$`y`=`s``i``n``x` and $y=\cos x$`y`=`c``o``s``x` have a maximum value of $1$1, a minimum value of $-1$−1, and an average value of $0$0. The distance of the maximum (or minimum) value from the average value is called the amplitude.

The range of sine and cosine functions is given by the difference between their maximum and minimum values. The amplitude is then half of the range (that is, $\frac{\text{max}-\text{min}}{2}$max−min2). Note that the amplitude is always a positive number.

The tangent function can obtain a value of any real number depending on the angle - that is, the tangent function varies between $+\infty$+∞ and $-\infty$−∞, and so there is no meaningful interpretation of amplitude for the tangent function.

In general, we say that the graph of a function $y=f\left(x\right)$`y`=`f`(`x`) is vertically translated when the resulting graph has the form $y=f\left(x\right)+c$`y`=`f`(`x`)+`c`, where $c$`c` is a non-zero constant.

Graphically speaking, a vertical translation takes each point on the graph of $y=f\left(x\right)$`y`=`f`(`x`) and adds (or subtracts) a constant to the $y$`y`-value of each point. For instance, all of the points on the graph of $y=\cos x$`y`=`c``o``s``x` shown below have been shifted upwards according to the constant term.

$y=\cos x$y=cosx vertically translated upwards by a positive constant $c$c. |

The constant term does not necessarily have to be positive. In the case that $c$`c` is negative (that is $c<0$`c`<0), the resulting graph will be translated vertically downwards.

$y=\sin x$y=sinx vertically translated downwards by a negative constant $c$c. |

Note that for the sine, cosine, and tangent functions, vertical translations do not change the domain. For sine and cosine functions the limits of the range either increase or decrease by the amount of the vertical translation, while the range of the tangent function remains unchanged when vertically translated.

Sketch the function $y=|2\sin x|$`y`=|2`s``i``n``x`| for $0\le x\le2\pi.$0≤`x`≤2π.

**Think**: First we should graph the function $y=2\sin x$`y`=2`s``i``n``x` without the absolute value signs.

**Do**: We know that the amplitude of this function is $2$2 and the period of the function is $2\pi.$2π.

By substituting a few values of $x$`x` into the function we can quickly see the shape it will make:

- When $x=0:\quad y=2\sin0=0$
`x`=0:`y`=2`s``i``n`0=0 - When $x=\frac{\pi}{2}:\quad y=2\sin\frac{\pi}{2}=2\times1=2$
`x`=π2:`y`=2`s``i``n`π2=2×1=2 - When $x=\pi:\quad y=2\sin\pi=2\times0=0$
`x`=π:`y`=2`s``i``n`π=2×0=0

By plotting these values we can sketch the function $y=2\sin x$`y`=2`s``i``n``x`:

To graph the absolute value function $y=|2\sin x|$`y`=|2`s``i``n``x`|, we must reflect the part of the curve that is below the $x$`x`-axis, by the $x$`x`-axis, to get:

Which of the following is the graph of $y=\sin x+4$`y`=`s``i``n``x`+4?

- Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Which of the following is the graph of $y=\tan x-1$`y`=`t``a``n``x`−1?

- Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

A vertical dilation of a function occurs when we multiply the function by a scale factor. That is, a function $kf\left(x\right)$`k``f`(`x`) will be vertically dilated by a factor of $k$`k`. Equations of the form $y=k\sin x$`y`=`k``s``i``n``x` or $y=k\cos x$`y`=`k``c``o``s``x` undergo vertical dilation by a factor $k$`k` and have an amplitude of $k$`k` units.

Graphically this transformation corresponds to **stretching** the graph of $\sin x$`s``i``n``x` or $\cos x$`c``o``s``x`in the vertical direction when $k>1$`k`>1, and **compressing** the graph of $\sin x$`s``i``n``x` or $\cos x$`c``o``s``x` in the vertical direction when $0`k`<1. This action of stretching and compressing the graph of a function is known as dilation. We can also **reflect** the graph across the $x$`x`-axis by multiplying the whole function by $-1$−1. The diagram below shows sine graphs with different amplitudes.

When the graphs of $\sin x$`s``i``n``x` or $\cos x$`c``o``s``x` are vertically dilated by a factor $k$`k`, the domain remains constant but the range of the function changes by the factor of the dilation.

We can summarise the effects of vertical dilations on the graphs of $y=\sin x$`y`=`s``i``n``x` or $y=\cos x$`y`=`c``o``s``x` and their amplitudes as follows:

Function | Dilation | Amplitude |
---|---|---|

$y=k\sin x$y=ksinx |
vertical stretch $|k|>1$| vertical compression $0<|k|<1$0<| horizontal reflection $k<0$ |
$|k|$|k| |

$y=k\cos x$y=kcosx |
vertical stretch $|k|>1$| vertical compression $0<|k|<1$0<| horizontal reflection $k<0$ |
$|k|$|k| |

State the amplitude of the function $f\left(x\right)=5\sin x$`f`(`x`)=5`s``i``n``x`.

**Think**: When we compare $f\left(x\right)$`f`(`x`) to the standard function $\sin x$`s``i``n``x` we can see that all the function values of $5\sin x$5`s``i``n``x` will be five times larger than all the function values of $\sin x$`s``i``n``x`. This means that the amplitude of $5\sin x$5`s``i``n``x` is also five times larger than the amplitude of $\sin x$`s``i``n``x`.

**Do**: The amplitude of $\sin x$`s``i``n``x` is $1$1, so the amplitude of $f\left(x\right)=5\sin x$`f`(`x`)=5`s``i``n``x` is $5\times1=5$5×1=5.

**Reflect**: We can obtain the graph of $f\left(x\right)=5\sin x$`f`(`x`)=5`s``i``n``x` by starting with the graph of $y=\sin x$`y`=`s``i``n``x` and applying a vertical dilation by a factor of $5$5.

Determine the equation of the graphed function given that it is of the form $y=a\sin x$`y`=`a``s``i``n``x` or $y=a\cos x$`y`=`a``c``o``s``x`.

Loading Graph...

Consider the function $y=-3\cos x$`y`=−3`c``o``s``x`.

What is the maximum value of the function?

What is the minimum value of the function?

What is the amplitude of the function?

Select the two transformations that are required to turn the graph of $y=\cos x$

`y`=`c``o``s``x`into the graph of $y=-3\cos x$`y`=−3`c``o``s``x`.Vertical dilation.

AHorizontal translation.

BReflection about the $x$

`x`-axis.CVertical translation.

DVertical dilation.

AHorizontal translation.

BReflection about the $x$

`x`-axis.CVertical translation.

D