# Trigonometric Graphs - Amplitude and Periodicity

#### Contents

## Period of Trigonometric Functions

From the definition of the basic trigonometric functions as $x$- and $y$-coordinates of points on a unit circle, we see that by going around the circle one complete time $($or an angle of $2\pi),$ we return to the same point and therefore to the same $x$- and $y$-coordinates. This can be extended for going around the circle any multiple of times $($or any angle that is a multiple of $2\pi).$

For example,

$0 = \sin (0) = \sin (0 + 2\pi) = \sin (0 + 2 \cdot 2\pi) = \cdots = \sin(0 + k \cdot 2\pi)$

for any integer $k$.

This shows the trigonometric functions are *repeating*. These functions are called **periodic**, and the **period** is the minimum interval it takes to capture an interval that when repeated over and over gives the complete function.

The periods of the basic trigonometric functions are as follows:

$\begin{array}{|c|c|} \hline \text{Function} & \text{Period}\\ \hline \sin (\theta) , \cos ( \theta ) & 2\pi\\ \hline \csc (\theta) , \sec ( \theta) & 2\pi \\ \hline \tan (\theta), \cot ( \theta ) & \pi\\ \hline \end{array}$

## Amplitude of Trigonometric Functions

As we have seen, trigonometric functions follow an alternating pattern between hills and valleys. The **amplitude** of a trigonometric function is half the distance from the highest point of the curve to the bottom point of the curve:

$\text{(Amplitude)} = \frac{ \text{(Maximum) - (minimum)} }{2}.$

For example, if we consider the graph of $y=\sin(x)$

the amplitude is equal to $\frac{ \text{(Maximum) - (minimum)} }{2} = \frac{ 1 - (-1)}{2} = \frac{2}{2} = 1$. Similarly, the graph of $y=\cos(x)$ also has amplitude 1.

The amplitudes of the basic trigonometric functions are as follows:

$\begin{array}{|c|c|} \hline \text{Function} & \text{Amplitude}\\ \hline \sin (\theta) , \cos( \theta ) &1 \\ \hline \csc (\theta) , \sec ( \theta) & \text{N/A} \\ \hline \tan (\theta), \cot ( \theta ) & \text{N/A} \\ \hline \end{array}$

In transformation of trigonometric graphs, we see that multiplying a trigonometric function by a constant changes the amplitude. For example, the amplitude of the graph $y = 3 \sin(x)$ is $3$.

## Problem Solving - Basic

What are the amplitude and period of the graph $y = 5 \sin(x) - 2?$

Since the trigonometric function $\sin(x)$ is multiplied by the constant $5$, the amplitude of the resulting graph is $5$. The result of the transformation is to shift the graph vertically by $-2$ in the $y$-direction and stretch the graph vertically by a factor of $5.$ Since the graph is not stretched horizontally, the period of the resulting graph is the same as the period of the function $\sin(x)$, or $2\pi$. $_\square$

What are the amplitude and period of the graph $y =-100\cos(1234\pi)?$

It doesn't matter whether it's $-100$ or $100;$ at the turning points, $y = \pm100.$ So the amplitude is 100, and the fundamental period is $\frac{2\pi}{1234\pi} = \frac{1}{617}.\ _\square$

The fundamental period of a sine function $f$ that passes through the origin is given to be $3\pi$ and its amplitude is 5. Construct $f(x).$

Since it passes through the origin, it must be of the form $f(x) = A \sin(kx)$ as $f(0) = 0$. Because its amplitude is 5, $f(x) = \pm 5 \sin(kx)$.

And because its fundamental period is $3\pi$, $\frac{2\pi}k = 3\pi \implies k = \frac23$. So $f(x) = \pm5 \sin\big( \frac23 x\big).\ _\square$

## Problem Solving - Intermediate

What are the fundamental period and amplitude of the function $f(x) =40\sin(2x) + 9\cos(2x)?$

By $R$ method, we have $f(x) = \sqrt{40^2 + 9^2} \sin(2x + \alpha ) = 41 \sin(2x+ \alpha)$ for some constant $\alpha$ independent of $x$. So the amplitude is 41 while the fundamental period is $\frac {2\pi}2 = {\pi}$. $_\square$

What is the period of the function $h(x) = \sin\big( |123x|\big)?$

A simple sketch of graph around the vicinity of $x = 0$ of $h(x) =\sin(123x) = \begin{cases} {\sin(123x) } \\ { -\sin(123x) } \end{cases}$ shows that it is not periodic. $_\square$

## Problem Solving - Advanced

What are the amplitude and fundamental period of the function $f(x) = 64\cos^7(x) - 112\cos^5(x) + 56\cos^5(x) - 7\cos(x)?$

By double angle formula and triple angle formula, we are able to obtain the fact that $f(x) = \cos(6x)$. Thus its amplitude is simply 1 and the fundamental period is $\frac 6 {2\pi} = \frac3{\pi}$. $_\square$

Note that we can also prove this using Chebyshev polynomials.

**Cite as:**Trigonometric Graphs - Amplitude and Periodicity.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/trigonometric-graphs-amplitude-and-periodicity/