# Sin, Cos and Tan Graphs

## Sin, Cos and Tan Graphs Revision

**Sin, Cos & Tan Graphs **

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

Specifically, the graphs of

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

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 past 360\degree and before 0\degree, it repeats this exact same shape which lasts for 360\degree.

This is shown below.

**Cosine Graphs**

The graph

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 before 0\degree and past 360\degree.

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

y=\tan x between -90\degree and 90\degree

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

Key things you need to understand about this graph:

- 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. However, this period repeats every 180\degree, unlike the previous graphs that are 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.

## Sin, Cos and Tan Graphs Example Questions

**Question 1:** On the same axes, plot the functions y=\sin(x) and y=\cos(x) between -180\degree and 180\degree.

**[4 marks]**

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 solid black line represents the \sin graph and the dotted line represents the \cos graph.

**Question 2:** Plot the function y=\tan(x) from -360\degree to 360\degree.

**[2 marks]**

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:

**Question 3:** Plot the function y=-\cos(x) between 0\degree and 360\degree.

**[2 marks]**

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.

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

**Question 4:** Plot the function y=\sin(x) and y=2\sin(x) between -180\degree and 180\degree.

**[4 marks]**

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