What you need to know

On top of understanding and being able to the functions of sin, cos, and tan – click here (https://mathsmadeeasy.co.uk/gcse-maths-revision/trigonometry-gcse-revision-and-worksheets/) if you aren’t sure what I’m talking about – you must also know what their graphs look like. Specifically, the graphs of

y=\sin x,\,\,\,\,y=\cos x,\,\,\,\,\text{and}\,\,\,\,y=\tan x.

You may be asked to draw graphs for any values of x (where x is in degrees), but fortunately, these graphs are periodic, which means that after a certain point, the graph follows a pattern and repeats itself over and over. In this topic, we’ll look at one period of each graph, and then see how it repeats.

On top of understanding and being able to the functions of sin, cos, and tan – click here (https://mathsmadeeasy.co.uk/gcse-maths-revision/trigonometry-gcse-revision-and-worksheets/) if you aren’t sure what I’m talking about – you must also know what their graphs look like. Specifically, the graphs of

 

y=\sin x,\,\,\,\,y=\cos x,\,\,\,\,\text{and}\,\,\,\,y=\tan x.

 

You may be asked to draw graphs for any values of x (where x is in degrees), but fortunately, these graphs are periodic, which means that after a certain point, the graph follows a pattern and repeats itself over and over. In this topic, we’ll look at one period of each graph, and then see how it repeats.

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.

 

As you can see, it crosses the axis every 180\degree, hits its minimum every 360\degree, and hits its maximum every 360\degree.

 

Secondly, the graph of \mathbf{y=\cos x} between 0\degree and 360\degree looks like the graph on the right. This might initially look quite different to the sin graph we saw, but first impressions aren’t everything. For example, you’ll notice that this graph also has a maximum of 1 and minimum of -1.

The similarities go further. As before, 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.

 

Looking closely, we can see that the graph of y=\cos x is just the sin graph translated left by \mathbf{90\degree}. So, rather than peaking at 90\degree, the cos graph peaks at 0\degree, and rather than crossing the axis at 180\degree, it crosses at 90\degree, and so on. This is useful, because it means we only have to remember one shape. This shape is called a sin wave, and is used for many things, from sending radio signals to forming the basis of the sounds many of the sounds produced by musical synthesisers.

 

Another consequence of the fact that the cos graph is just the sin graph shifted left by 90\degree, is that, by knowing our transformations, we can write

 

\sin(x+90)=\cos(x).

 

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.

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

 

Furthermore, you can see that there is an asymptote every 180\degree also, but not in the same place that graph crosses the axis.

 

You may also be asked to transform these graphs, i.e. apply translations or reflections to any of the 3 functions that we saw here. As long as you know the graphs, the process of transforming is exactly the same as with any other graphs.

Now, the graph of \mathbf{y=\tan x} is very different. Between -90\degree and 90\degree, it looks like the graph on the right.

 

As you can see, it crosses the axes once at the origin, 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.

Example Questions

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

Answer

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:

 

 

2) Plot the function y=\tan(x) from 0\degree to 360\degree.

Answer

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:

 

 

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

Answer

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

 

 

If you are scouring the internet to find more sin, cos and tan revision materials for AQA, OCR and Edexcel, well you have stumbled upon the right website. At Maths Made Easy we produce high quality GCSE Maths revision materials for students, tutors and teachers to use. So whether you are a student in London or a GCSE Maths tutor in Leeds, you should find our sin, cos and tan resources really useful.

Where Next?