# Sine graph period and amplitude relationship

### 1. Graphs of y = a sin x and y = a cos x

Amplitude, Period, Phase Shift and Frequency. Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions. The Period goes from. Calculate the amplitude and period of a sine or cosine curve. .. you were given the frequency and asked to find the period using the following relationship. A sinusoid is a function of time having the following peak amplitude ω Figure 2: Phase relationship between cosine and sine functions.

## Amplitude, Period, Phase Shift and Frequency

At first glance they may look to be the same. But the graph on the right is a vertical shift of the graph on the left.

While vertical shifts alter the maximum and minimum values of a function, they do not alter the amplitude. Also, horizontal shifts do not affect the amplitude. A periodic function is a trigonometric function which repeats a pattern of y-values at regular intervals.

One complete repetition of the pattern is called a cycle. The period of a function is the horizontal length of one complete cycle. The period may also be described as the distance from one "peak" max to the next "peak" max.

In functional notation we could say: The frequency of a trigonometric function is the number of cycles it completes in a given interval. In terms of a formula: It is also true that: In application problems, when frequency is measured "per second" it is called "Hertz" 10 Hertz means 10 times per second. Let us first consider the shape of the function Since our original function, is a periodic function that goes through one complete cycle whenthe function will go through a complete cycle wheni.

We say that it has a period which we will denote by given by The height of the peaks and valleys in this function will be given by its amplitude.

### Amplitude, Period, Phase Shift and Frequency

We are now ready to consider the effect of the phase-shift. In fact, we can make note of the fact that the graph of the function will cross the t axis when The first time that this happens is when which corresponds to a value of t given by Thus, the graph will be shifted so that it crosses the t axis at this value.

The shape of the curve does not change, only its position on the t axis. Superimposing sines and cosines Let us take a second look at the function we investigated above, and notice that when we apply the trigonometric identity we obtain is a constant, and therefore so isand assigning the names we have found that Thus, by using a trigonometric identity for the sums of angles, we have reduced a problem we needed to understand the question we started with, at the top of this page with a problem that we already know how to solve.

We have found that the sum of a sine and a cosine curve is actually equivalent to a sine with a phase shift.

A bit of care is required, however, since in order for this conversion to work, it must be true that For your consideration: Describe the behaviour of the function Solution: We observe that the constants in front of the trigonometric functions have the values We would like to find the angle and the amplitude that fit with this pattern.

The ratio of the constants Thus, looking up the angle that has a value of we find that Thus the phase shift is.