## FG10 (T8) Graphs of sine and cosine functions

Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians.

The functions $$y=\sin x$$ and $$y=\cos x$$ have a domain of $$\mathbb{R}$$ and a range of $$[-1,1]$$.

The graphs of these functions are periodic graphs, that is, the shape of the graph repeats every set period.

The graphs of both functions have an amplitude of $$1$$ and a period of $$2\pi$$ radians (that is the graph repeats every $$2\pi$$ units). They are shown below.

When looking at the graphs remember $$\pi\approx3.142,$$ so $$2\pi\approx6.284$$.

In this module we look at how the basic graphs may be transformed into graphs of more complex trigonometric functions.

### Change of Amplitude and Period

The graphs of both $$y=a\sin nx$$ and $$y=a\cos nx$$ have an amplitude $$\left|a\right|$$ and a period of $$\frac{2\pi}{n}$$.

#### Examples

1. Graph $$y=3\sin x$$.

In this case, $$a=3$$ and $$n=1$$, therefore the graph has an amplitude of $$3$$ and period of $$2\pi$$.

1. Graph $$y=3\cos2x$$.

In this case, $$a=3$$ and $$n=2$$, therefore the graph has an amplitude of $$3$$ and period of $$\frac{2\pi}{2}=\pi$$.

### Vertical translation

The graph of $$y=a\sin nx+k$$ is the graph of $$y=a\sin nx$$ translated up $$k$$ units (or down $$k$$ units if $$k$$ is negative).

The graphs of $${\color{blue}y=\sin x+2}$$ and $${\color{red}y=\sin x}$$ are shown below.

Similarly, the graph of $$y=a\cos nx+k$$ is the graph of $$y=a\cos nx$$ translated up $$k$$ units (or down $$k$$ units if $$k$$ is negative).

### Horizontal Translation

Replacing the $$x$$ with $$\left(x-\phi\right)$$ shifts the graphs of $$y=\sin x$$ and $$y=\cos x$$ horizontally $$\phi$$ units to the right.

Replacing the $$x$$ with $$\left(x+\phi\right)$$ shifts the graphs of $$y=\sin x$$ and $$y=\cos x$$ horizontally $$\phi$$ units to the left.

#### Examples

1. Graph $$y=\sin\left(x-\frac{\pi}{2}\right)$$

The graph of $$y=\sin\left(x-\frac{\pi}{2}\right)$$ shown in blue, superimposed on the graph of $$y=\sin x,$$ in dashed red is shown below.

1. Graph $$y=\cos\left(x+\pi\right)$$

The graph of $$y=\cos\left(x+\pi\right)$$, shown in blue, superimposed on the graph of $$y=\cos x$$, in dashed red, is shown below.

1. Graph $$y=3\sin\left(4x-\pi\right)$$

1 First change $$y=3\sin\left(4x-\pi\right)$$ to the form $$y=3\sin4\left(x-\frac{\pi}{4}\right)$$ so that the horizontal translation of the graph is clear.

The graph of $$y=3\sin4\left(x-\frac{\pi}{4}\right)$$ in black is superimposed on the graphs of $$y=3\sin x$$ (dotted red) and $$y=3\sin4x$$ (dashed grey).

### Reflection

Changing the sign of $$a$$ in the equations $$y=a\sin nx$$ and $$y=a\cos nx$$ results in reflection about the $$x$$-axis.

#### Example

Graph $$y=-3\cos2x$$.

The graph of $$y=-3\cos2x$$ (in black) superimposed on the graph of $$y=3\cos2x$$ (dotted).

### Exercise 1

1. Sketch the graphs of the following functions for one complete cycle stating the amplitude and the period.

(a)$$\,$$ $$y=2\cos x$$

(b)$$\,$$ $$y=2\sin3x$$

(c)$$\,$$ $$y=\frac{1}{2}\sin2x$$

(d)$$\,$$ $$y=3\cos\frac{x}{2}$$

(e)$$\,$$ $$y=-2\sin3x$$

1(a)

Amplitude = $$2$$ , Period = $$2\pi$$

1(b)

Amplitude = $$2$$ , Period = $$\dfrac{2\pi}{3}$$

1(c)

Amplitude = $$\dfrac{1}{2}$$ , Period = $$\pi$$

1(d)

Amplitude = $$3$$ , Period = $$4$$ $$\pi$$

1(e)

Amplitude = $$2$$ , Period = $$\dfrac{2\pi}{3}$$

### Exercise 2

Sketch the graphs of the following functions for one complete cycle stating the amplitude and period.

1. $$y=2\sin\left(x-\pi\right)$$

2. $$y=3\cos\left(x+\frac{\pi}{2}\right)$$

2(a)

Amplitude = $$2$$ , Period = $$2\pi$$

2(b)

Amplitude = $$3$$ , Period = $$2\pi$$

### Exercise 3

Sketch the graphs of the following functions for one complete cycle stating the amplitude and period.

1. $$y=2\sin\left(3x-\pi\right)$$

2. $$y=3\cos\left(4x-2\pi\right)$$

3. $$y=2\sin\left(2x+\frac{\pi}{3}\right)$$

3(a)

Amplitude = $$2$$ , Period = $$\dfrac{2\pi}{3}$$

3(b)

Amplitude = $$3$$ , Period = $$\dfrac{\pi}{2}$$

3(c)

Amplitude = $$2$$ , Period = $$\pi$$

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