Double-angle identities

[samlipsky]

Cos x= Cos 2x What are the values of x between -360 and 360? Answer using double angle identities:

Cos 2x = 2(cos^2)x - 1
or
Cos 2x = 1 - 2 (sin^2)x

 

[soroban]

Hello, samlipsky!

$\displaystyle \cos x \:= \:\cos 2x$
What are the values of $\displaystyle x$ between -360° and 360°?

Answer using double angle identities: $\displaystyle \:\begin{array}{cc}\cos 2x \:= \:2\cos^2x\,-\,1 \\ \cos2x \:= \:1\,-\,2\sin^2x\end{array}$

Use the first identity and get the equation in terms of $\displaystyle \cos x$

We have: $\displaystyle \:\cos x\:=\:2\cos^2x\,-\,1$

Then: $\displaystyle \:2\cos^2x\,-\,\cos x \,-\,1\;=\;0$

. . which factors: \(\displaystyle \\cos x\,-\,1)(2\cos x\,+\,1)\;=\;0\)


And we have two equations to solve:

. . \(\displaystyle \cos x\,-\,1\:=\:0\;\;\Rightarrow\;\;\cos x\:=\:1\;\;\Rightarrow\;\;\fbox{x\:=\:-360^o,\:0^o,\:360^o}\)

. . \(\displaystyle 2\cos x\,+\,1\:=\:0\;\;\Rightarrow\;\;\cos x\:=\:-\frac{1}{2}\;\;\Rightarrow\;\;\fbox{x\:=\:-240^o,\:-120^o,\:120^o,\:240^o}\)