Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {43}In general, the same method will work to add a sin x +b cos x, in which case we divide outby a 2 + b 2 .Exercise 19Express 3cos x − 4sin x in the form A sin(x + α).Waves such as the square wave and the saw-tooth wave, which arise in physics and engineering,can be approximated using the sum of a large number of waves — this is thestudy of Fourier series, which is central to much of modern electrical engineering andtechnology.The following square wave is the graph of the function with rule12 + 2 π cos x − 23π cos3x + 25π cos5x − ··· = 1 2 + 2 π∞∑n=1sin nπ2ncosnx.y0xThe t formulasIntroducing the parameter t = tan θ turns out to be a very useful tool in solving certain2types of trigonometric equations and also in finding certain integrals involving trigonometricfunctions. The basic idea is to relate sinθ, cosθ and even tanθ to the tangent ofhalf the angle. This can be done using the double angle formulas.We let t = tan θ 2 and so we can draw the following triangle with sides 1, t, 1 + t 2 .√1 + t 2tθ21