This problem introduces students to different definitions or representations of trigonometric functions and challenges them to consider when each different representation is most useful.
Start by inviting students to discuss in pairs how they would define $\sin x$ and $\cos x$, and then share these discussions as a group. It is likely that they will consider trigonometric functions defined in terms of right-angled triangles. (Note that the diagram in the problem gives a slightly unusual representation of this definition- this representation hints at double angle formulae and
calculus and might itself provoke some discussion).
Introduce the two other representations from the problem. The representations are all available on this worksheet .
Next, invite students to think of as many properties of sin and cos that they can think of, and list them on the board. This worksheet lists the properties given in the problem, and includes some blank rows for any the students think of for themselves that they wish to work on.
"For each property, use the three representations to see if you can deduce the property. It might be easier in some representations than others, and it might be impossible in some representations! Make a note in each cell of the table whether it was easy, difficult or impossible to prove."
Give students time to work on the task in pairs or small groups. As they are working, circulate and listen for clear explanations or insights about the benefits and limitations of the three representations. Finally, invite students out to the board to explain their proofs for each property and to comment on the three representations.
Which representation makes it easy to prove this?
Is it possible to prove this using all three representations?
Students who readily solve this challenge could be given the power series and exponential definitions of sinh and cosh and asked to reconstruct a table of properties of sinh and cosh by analogy. This extension would work well if encountered before formally meeting sinh and cosh. Those students who have met sinh and cosh might additionally be asked to construct an analogue of the first trig rep
for the hyperbolic functions.
The animation in Round and Round a Circle could be used as an introduction for students whose understanding of the right-angled triangle representation is not secure.