Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {41}y y = tan x3210–2 –3 – – 3 22 222–1–2–3xThe period of the graph of y = tannx is given by π n .Links forwardMultiple anglesThe double angle formulas can be extended to larger multiples. For example, to findcos3θ, we write 3θ as 2θ + θ and expand:cos3θ = cos(2θ + θ) = cos2θ cosθ − sin2θ sinθ.We can now apply the double angle formulas to obtaincos3θ = ( cos 2 θ − sin 2 θ ) cosθ − 2sinθ cosθ sinθ = cos 3 θ − 3sin 2 θ cosθ.Replacing sin 2 θ with 1 − cos 2 θ, we havecos3θ = 4cos 3 θ − 3cosθ.Exercise 18Use the method above to find a formula for sin3θ.A more general approach is obtained using complex numbers. The complex number i isdefined by i 2 = −1.De Moivre’s theorem says that, if n is a positive integer, then( ) n cosθ + i sinθ = cosnθ + i sinnθ.Hence, for any given n, we can expand (cosθ + i sinθ) n using the binomial theorem andequate the real and imaginary parts to find formulas for cosnθ and sinnθ.