Sin, Cos and Tan Graphs Worksheets | Questions and Revision | MME

Sin, Cos and Tan Graphs Worksheets, Questions and Revision

Level 6 Level 7

What you need to know

Sine, cosine and tangent graphs are specific graphs you need to be able to identify, understand and draw.

Specifically, the graphs of

$y=\textcolor{red}{\sin} x,\,\,\,\,y=\textcolor{blue}{\cos} x,\,\,\,\,\text{and}\,\,\,\,y=\textcolor{limegreen}{\tan} x$.

You may be asked to draw graphs for any values of $x\degree$, these graphs are periodic, which means that after a certain point, the graph follows a pattern and repeats itself over and over.

Make sure you are happy with the following topics before moving on:

Sine graphs

The graph

$\mathbf{y=\sin x}$ between $0\degree$ and $360\degree$

is the graph shown below.

The key features of this graph you need to remember

• The peak is $\mathbf{1}$ and occurs at $\mathbf{90}\degree$,
• The minimum is $\mathbf{-1}$ and occurs at $\mathbf{270}\degree$
• The graph crosses the axis at $\mathbf{0}\degree, \mathbf{180}\degree,$ and $\mathbf{360}\degree$.

As mentioned, this is one period, which means that above $360\degree$ and below $0\degree$, it repeats this exact same shape which lasts for $360\degree$.

This is shown below.

Cosine graphs

The graph

$\mathbf{y=\cos x}$ between $0\degree$ and $360\degree$

is shown below.

The key features of this graph you need to remember

• The peak is $\mathbf{1}$ and occurs at $\mathbf{0}\degree$ and $\mathbf{360}\degree$,
• The minimum is $\mathbf{-1}$ and occurs at $\mathbf{180}\degree$
• The graph crosses the axis at $\mathbf{90}\degree$ and $\mathbf{270}\degree$.

As with the sine graph, this portion is one period of the graph, so it is repeated for all the values below $0\degree$ and above $360\degree$.

If we repeat this period a few times, we will see that the shape is exactly the same as the sin graph.

NOTE: if, at any point, you can remember the general shape of these graphs but can’t remember which graph is which, you can recall/calculate the values of sin and cos at zero, and then extend the pattern from there onward.

Tangent graphs

The graph

$\mathbf{y=\tan x}$ between $-90\degree$ and $90\degree$

is very different, it looks like the graph shown below.

• It crosses the axes once at the origin
• The graph gets very big as the angle gets close to $90\degree$, and similarly gets very small as the angle gets close to $-90\degree$.
• The dotted lines on this graph are asymptotes – lines which the function gets closer and closer to but never quite touches.

As with the previous graphs, this part only represents one period. This period repeats every $180\degree$ however, unlike the previous graphs that repeated every $360\degree$. Note: as the graph repeats, so do the asymptotes.

The result of repeating the shape a few times is shown below.

If anything, this graph is slightly simpler than the previous two, because it only crosses the axis once every $180\degree$.

Example Questions

If you can’t remember their shapes, check a few points. So, we have that

$\cos(0)=1,\,\,\text{ and }\,\,\cos(90)=0$

Which is enough to start of the pattern of the cos graph. Similarly, we have

$\sin(0)=0,\,\,\text{ and }\,\,\sin(90)=1$

Which is enough to start the pattern of the sin graph. If you aren’t sure, just try more values. The resulting graph looks like:

The tan graph has an asymptote at $90\degree$, and then again every $180\degree$ before and after that. Furthermore, we have that $\tan(0)=0$ and it gets bigger as it gets close to $90\degree$. This enough to draw the graph. The result looks like:

This is a transformation of the form $y=-f(x)$, which corresponds to a reflection in the $x$ axis. In doing this, it would be helpful for you to draw a normal cos graph, draw the reflection, and then rub out the first one. Here, we’re going to show the normal cos graph as a dotted line.

To draw the cos graph, consider that

$\cos(0)=1\,\,\text{ and }\,\,\cos(90)=0$

This is enough to continue the pattern to $360\degree$. The resulting graph should look like

Here we will plot $y=\sin(x)$ as a dotted line and $y=2\sin(x)$ as a solid line. The resulting graph should look like

The resulting graph should look like:

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