Why do I prefer solving with graphs as opposed to the CAST diagram?

Some A-level students across the country are taught to solve trig equations through a CAST diagram. In fact I was also one of them!

I had diligently applied the CAST diagrams to many trig problems in my youth, however when I started teaching it to Year 12 students in the early part of my career I suddenly understood its limitations.

Those Year 12 students who entered into Year 13 with a different teacher and in a mixed class were in a minority and got found out when the rest of the class were taught to solve trig equations graphically.

After a period of reflection I understood the limitations of the CAST diagram:

• Although it is linked to the trig graphs themselves I found many students struggled to get their heads around it because its not explicitly linked in the textbooks.

• As a teacher I also found the students who did like the CAST diagram often had poor graph sketching and transforming skills which invariably hindered them when they went into year 13.

• If the trig question had a weird range to solve in some students really found CAST quite difficult to use.

The case for solving by sketching graphs

Example of solving trig equations by graphical method

Look at the questions in the above image, you can clearly see that by sketching the graph of sin(x) for the range desired you can solve if you draw the horizontal line of 0.2 or -0.2.

This is still the same even if the range was a little more difficult, say between -270 and 540 degrees. You can extend the sine graph and find more solutions.

What are the benefits of the graphical approach?

• You embed the graphs into your long term memory if you practice solving in this manner. This helps you remember key facets of each graph.

• Your graph sketching ability improves and this is a key skill in general.

• You can edit the graph by taking it as far as you need for different ranges.

• Your ability to deal with transformations improve as you are able to recall the trig graphs in detail and can apply different transformations to them without hassle.

Of course this is just an opinion piece based on my experience. I was taught CAST exclusively and tried to teach it at A-level but over time I felt the graphical method paid dividends. If you are a die-hard CAST fan then carry on using it, but do not discount the benefits of solving with the graphical method!