Problem 2

Let

a_1,a_2,\ldots,a_n

be real constants and let

x

be real. Define

f(x)=\cos(a_1+x)+\frac12\cos(a_2+x)+\cdots+\frac1{2^{n-1}}\cos(a_n+x)

. Given that

f(x_1)=f(x_2)=0

, prove that

x_2-x_1=m\pi

for some integer

m

Step 3 of 4: Write one phase-shifted cosine

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In plain words

A linear combination of sine and cosine with one frequency is just a cosine shifted horizontally.

R=\sqrt{B^2+C^2}>0,\qquad f(x)=R\cos(x+\delta)

Detailed analysis

Choose $\delta$ with $\cos\delta=B/R$ and $\sin\delta=C/R$ . Then $R\cos(x+\delta)=B\cos x-C\sin x=f(x)$ , and the amplitude $R$ is positive by the preceding step.