Why use this resource?
Thinking about the behaviour of functions can help to make sense of identities. Similarly, by looking at identities satisfied by a function we can glean information about the behaviour of the function. In this resource, students are asked to use the half-angle formulae \(\cos^2 \frac{\theta}{2} \equiv \frac{1}{2}(1+\cos \theta)\) and \(\sin^2 \frac{\theta}{2} \equiv \frac{1}{2}(1-\cos \theta)\) to help them sketch graphs of \(y=\cos^2{\frac{\theta}{2}}\) and \(y=\sin^{2}\frac{\theta}{2}\).
Preparation
If printed graphs of \(y=\cos \theta\) are available these could be used for students to sketch onto. Otherwise, \(\quantity{1}{cm}\) square paper would be useful.
Possible approach
Students could be asked to start by sketching a \(y=\cos \theta\) graph if pre-printed graphs are not available.
Ask students to think individually about what they would expect a \(y=\cos^2 \frac{\theta}{2}\) curve to look like using their cosine graph sketch as a starting point. After a minute or so they should add their \(y=\cos^2 \frac{\theta}{2}\) curve to their sketch then discuss their ideas with a partner. Pairs can then form small groups to share their ideas and agree on what they think the essential features of \(y=\cos^2 \frac{\theta}{2}\) and \(y=\cos^2 \frac{\theta}{2}\) are.
If graphing software is available students could check their graphs or this could be done as a whole class plenary, asking students to explain why the graphs have the features shown.
Key questions
- Are there any points on the graph we know straight away?
- What are the domain and range of the two functions?
- Can you describe \(y=\cos^2 \frac{\theta}{2}\) and \(y=\sin^2 \frac{\theta}{2}\) in terms of transformations of \(y=\cos \theta\)?
Possible support
To support students in thinking about functions and their graphs
- How could you decide whether \(\cos^2 \frac{\theta}{2}\) and \(\cos \theta\) have the same period?
- What is the range of \(\sin^2 \frac{\theta}{2}\) and \(\cos \theta\)?
To support students in thinking about graph transformtions when looking at \(\frac{1}{2}(1+\cos \theta)\) and \(\frac{1}{2}(1-\cos \theta)\)
- How does the \(+1\) transform the graph?
- What effect does the minus sign before the cosine function have?
- What effect does the coefficient of \(\frac{1}{2}\) have on the cosine graph?
Possible extension
- Can you transform a graph of \(y=\sin \theta\) into either of these half-angle graphs?
- Can you use that to write down a new identity?
A version of this resource has been featured on the NRICH website.
