Consider the following points on the graphs of \(y = \cos{x}\) and \(y = \cos{\frac{x}{2}}\) with the added vertical dashed lines:

Find C and D

I tried finding the cosine of points a and b but not sure how I continue here

YourAverageDummy Jan 30, 2023

#8**+1 **

I think there are a couple of different ways to take lthis question.

Looking at the graph I can see that b=a+2pi

From a to b is a whole wavelength of cos a and a half a wavelength of cos (a/2)

This means that

1-d = c - - 1

1-d= c+1

c=-d

\(cos\;a= \frac{1}{8}\\ cos(\frac{a}{2}+\frac{a}{2})=\frac{1}{8}\\ cos^2(\frac{a}{2})-sin^2(\frac{a}{2})=\frac{1}{8}\\ cos^2(\frac{a}{2})-(1-cos^2(\frac{a}{2}))=\frac{1}{8}\\ 2cos^2(\frac{a}{2})-1=\frac{1}{8}\\ 2cos^2(\frac{a}{2})=\frac{9}{8}\\ cos^2(\frac{a}{2})=\frac{9}{16}\\ cos(\frac{a}{2})=\pm\frac{3}{4}\\ \text{discount the positive answer}\\ c=cos(\frac{a}{2})=-\frac{3}{4}\\ \)

\(d=\frac{3}{4}\)

Melody Jan 31, 2023

#9**+1 **

huh. I never thought that you could prove c in terms of d

Thanks Melody!

YourAverageDummy Jan 31, 2023