# Compound angles proof

by BOAS
Tags: angles, compound, proof
PF Gold
P: 215
Hello,

simple question.

My textbook (Bostock and Chandler - Pure Mathematics 1) says something that really surprises me.

 When the same investigation is carried out on $f(\theta)$ $\equiv$ $sin3\theta$ we find that the function is cyclic with a period of $\frac{2\pi}{3}$ so that $3$ complete cycles occur between $0$ and $2\pi$. It seems likely (Although it has not been generally proved) that the graph of the function $f(\theta)$ $\equiv$ $sink\theta$ is a sine wave with a period of $\frac{2\pi}{k}$ and a frequency $k$ times that of $f(\theta)$ $\equiv$ $sin\theta$
The bolded part is what shocked me, it seems like such a trivial statement and intuitively true. My book was first published in 1978, so perhaps it is out of date.

It goes on to say;

 These properties are, in fact, valid for all values of k
Which seems contradictory... So, has or has not this idea been proven true?

Thanks!
 Sci Advisor P: 5,935 It is trivially obvious, why are you puzzled? What is contradictory? I presume the statement is for k a positive integer.
PF Gold
P: 215
 Quote by mathman It is trivially obvious, why are you puzzled? What is contradictory? I presume the statement is for k a positive integer.
I mean that it seems trivial, so I was surprised that it had not been proven true. By contradictory, I mean, the book says the idea is not generally proven but goes on to say that it is true for all values of k.

It is not explicitly stated in my textbook what is meant by k, but all related questions deal with positive numbers, fractions and integers.