Trig Identity

Why do this problem?

This problem offers students the opportunity to find expressions for $\cos 2 A$ and $\sin 2 A$.  They have to work out what they know, and what they can deduce given the information they have.

Students will need to bring together that they know about circle theorems, Pythagoras' theorem and properties of isosceles triangles to find the identities for $\cos 2A$ and $\sin 2A$.

Here are printable versions of the problem in Word and PDF format.  They include extra diagrams for students to annotate.

Possible approach

Start with the diagram on the board and ask students if there are any angles or lengths they can put on the diagram given the information from the question.  

Once the information from the question is on the diagram, ask students if there any other angles or lengths they can deduce.  Ask them if they can find or create any right angled triangles.

Once the diagram has been adapted to show the lengths/angles that were known or could be deduced asked students if they can find a way to include $\sin 2A$ and/or $\cos 2A$ in a relationship involving a right-angled triangle.

Key questions

• What information are we given in the question?
• How we can display the information from the question on the diagram?
• What other angles can we deduce?
• Is there anything special about the two triangles on the diagram?
• What do you know about isosceles triangles?
• How many right-angled triangles can you find?

Possible extension

Students might like to explore T for tan which uses similar ideas to find some more trig identities.