5.3 – Trigonometric Graphs

5.3 Video 1 of 3:Sine and Cosine curves

5.3 Video 2 of 3:Amplitude, Period and Phase Shift

5.3 Video 3 of 3:Trig Graph Examples

Section Notes

Trigonometric Graphs (Sin and Cos)

Viewing the graph of a function can give insight to its behavior. This section will examine the graphs of the trigonometric functions.

We first want to note that, as we are dealing with a unit circle, when $t$ circumnavigates this circle, the sine and cosine values repeat themselves. This is known as periodicity. We say that sine and cosine are periodic functions. This can be stated mathematically:

Periodic Properties of Sine and Cosine
Sine and Cosine both have period $2\pi$ , thus: $\cos(t+2\pi)=\cos(t) \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \sin(t+2\pi)=\sin(t)$

Periodic Properties of Sine and Cosine

Sine and Cosine both have period $2\pi$ , thus:

$\cos(t+2\pi)=\cos(t) \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \sin(t+2\pi)=\sin(t)$

Because the functions repeat themselves every $2\pi$ , we can look at one period to see the behavior of the graph. We will do this by taking $t$ as the horizontal axis of our graph, and the function as the vertical. First, let us look at the function $\sin t$ , and begin by listing out the known values:

$\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline$\textbf{t}$ & \raisebox{0em}[1em][1em]{$0$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{6}$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{4}$} &\raisebox{0em}[2em][1em]{$\dfrac{\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{2\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{3\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{6}$} & $\pi$ & \raisebox{0em}[2em][1em]{$\dfrac{7\pi}{6}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{4\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{3\pi}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{7\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{11\pi}{6}$} & $2\pi$\\ \hline $\textbf{Sin t}$ & \raisebox{0em}[1em][1em]{$0$} & \raisebox{0em}[2em][1em]$\dfrac{1}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$1$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{1}{2}$} & $0$ & \raisebox{0em}[2em][1em]{$-\dfrac{1}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$-1$} & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{1}{2}$} & $0$ \\ \hline\end{tabular}$ Graphing these values and connecting them with a smooth curve, we can see the shape of the function $\sin t$ :

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Using the same method, we can find the points of $\cos t$ , graph them, connecting them with a smooth curve through one period:

$\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline$\textbf{t}$ & \raisebox{0em}[1em][1em]{$0$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{6}$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{4}$} &\raisebox{0em}[2em][1em]{$\dfrac{\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{\pi}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{2\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{3\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{6}$} & $\pi$ & \raisebox{0em}[2em][1em]{$\dfrac{7\pi}{6}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{4\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{3\pi}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{5\pi}{3}$} & \raisebox{0em}[2em][1em]{$\dfrac{7\pi}{4}$} & \raisebox{0em}[2em][1em]{$\dfrac{11\pi}{6}$} & $2\pi$\\ \hline $\textbf{Cos t}$ & \raisebox{0em}[1em][1em]{$1$} & \raisebox{0em}[2em][1em]$\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{1}{2}$} & \raisebox{0em}[2em][1em]{$0$} & \raisebox{0em}[2em][1em]{$\dfrac{-1}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{-\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{-\sqrt{3}}{2}$} & $-1$ & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{3}}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$-\dfrac{1}{2}$} & \raisebox{0em}[2em][1em]{$0$} & \raisebox{0em}[2em][1em]{$\dfrac{1}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{2}}{2}$} & \raisebox{0em}[2em][1em]{$\dfrac{\sqrt{3}}{2}$} & $1$ \\ \hline\end{tabular}$

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The graphs above are just one period of each of the functions. They continue forever in each direction in a similar fashion. Let us expand these functions further:

The graph of $\sin t$ :

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And the graph of $\cos t$ :

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Trigonometric graphs can be shifted up, down, left or right by quick manipulations. To shift the graph of $y=\cos t$ up by 3 units, 3 is simply added to the function. That is, the function $y=3+\cos t$ is the graph of $y=\cos t$ shifted up by 3 units:

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The graph of $y=\sin t$ can also be “stretched” vertically by multiplying by a constant number. Shown below are a few multiples of the Sine function:

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And of course, these functions can be multiplied by a negative constant number. This will reflect the function across the horizontal axis. Here are a few graphs of the Cosine function and the Cosine function multiplied by various negative numbers:

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Trigonometric functions can also have their periods altered. Below is the graph of $y=\sin t$ and $y=\sin 2t$ on a single set of axes:

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You might notice that the period of the function $y=\sin 2t$ is smaller than the period of $y=\sin t$ . In fact, there are two periods of $y=\sin 2t$ in the same $2\pi$ interval that holds one period of $y=\sin t$ .

Below is graphed one period of an assortment of different periodic Sine functions:

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Another way to manipulate these graphs is a horizontal shift. This can be accomplished by adding or subtracting a number within the trigonometric function itself. The graph of $y=\cos (t-1)$ is the graph of $y=\cos t$ shifted to the right by 1 unit. the graph of $y=\cos (t+1)$ is the graph of $y=\cos t$ shifted to the left by 1 unit.

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These graph manipulations can be put together to describe the shape of many Sine and Cosine functions:

Sin and Cosine Graphs
The sine and cosine curves $y=a\sin[b(t-c)] +d \ \ \ \ \ \ \ and \ \ \ \ \ \ \ y=a\cos[b(t-c)]+d$ have an amplitude of $\|a\|$ , period of $\frac{2\pi}{b}$ , are horizontally shifted by $c$ units, and vertically shifted by $d$ units. One period is graphed between the interval $\left[c,\frac{2\pi}{b}+c\right]$

Sin and Cosine Graphs

The sine and cosine curves

$y=a\sin[b(t-c)] +d \ \ \ \ \ \ \ and \ \ \ \ \ \ \ y=a\cos[b(t-c)]+d$

have an amplitude of $|a|$ , period of $\frac{2\pi}{b}$ , are horizontally shifted by $c$ units, and vertically shifted by $d$ units.

One period is graphed between the interval $\left[c,\frac{2\pi}{b}+c\right]$

Find the amplitude, period, horizontal and vertical shifts and then graph the function $y=3\cos[4(t-3)]+2$
This graph will have an amplitude of 3, a period of $\frac{2\pi}{4}=\frac{\pi}{2}$ , a horizontal shift of 3 and a vertical shift of 2. One period is graphed in the interval $\left[3,\frac{\pi}{2}+3\right]$ : And to show this graph over multiple periods, it is simply extended to the left and the right using the same periodic length:

Find the amplitude, period, horizontal and vertical shifts and then graph the function $y=3\cos[4(t-3)]+2$

This graph will have an amplitude of 3, a period of $\frac{2\pi}{4}=\frac{\pi}{2}$ , a horizontal shift of 3 and a vertical shift of 2. One period is graphed in the interval $\left[3,\frac{\pi}{2}+3\right]$ :

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And to show this graph over multiple periods, it is simply extended to the left and the right using the same periodic length:

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PreCalculus I

5.3 – Trigonometric Graphs

5.3 Video 1 of 3:Sine and Cosine curves

5.3 Video 2 of 3:Amplitude, Period and Phase Shift

5.3 Video 3 of 3:Trig Graph Examples

Section Notes

Trigonometric Graphs (Sin and Cos)

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