# Difference between $\cos 2\theta$ and $\cos \theta/2$

I would like to know the difference between $\cos 2\theta$ and $\cos \theta/2$. Lets say cos $\theta = 1/2$. The answer would be $60$ and $300$. SO would the answer for $\cos 2\theta = 1/2$ be $420$ and $660$? I say that because I assume that $2\theta$ would double the original values.

-

The solutions of $\cos \theta=\mathrm{smth}$ are defined up to addition of $360^{\circ}$. So if you find the solutions of $\cos \theta=\frac12$ to be $60^{\circ}$ and $300^{\circ}$ (which is correct), you can still add to them any integer multiple of $360^{\circ}$.

Now if you have the equation $\cos 2\theta=\frac12$, the solutions will look like $$2\theta=60^{\circ}\text{ or } 300^{\circ}\; +n\cdot 360^{\circ},$$ which implies $$\theta=30^{\circ}\text{ or } 150^{\circ}\; +n\cdot 180^{\circ},$$ and therefore you will get four solutions between $0^{\circ}$ and $360^{\circ}$, namely: $$30^{\circ}, 150^{\circ}, 210^{\circ}\text{ and }330^{\circ}.$$

Added: Similarly, for $\cos \frac{\theta}{2}=\frac12$, we start by writing $$\frac{\theta}{2}=60^{\circ}\text{ or } 300^{\circ}\; +n\cdot 360^{\circ},$$ which then gives $$\theta=120^{\circ}\text{ or } 600^{\circ}\; +n\cdot 720^{\circ},$$

-
Thank you for your help but what about cos θ/2? How does that affect the result? –  Average_Ted Oct 13 '13 at 13:56
@Average_Ted Then, similarly you write $\theta/2 = 60^{\circ}$ or $300^{\circ}$ plus $n\cdot 360^{\circ}$, which implies that $\theta=120^{\circ}$ or $600^{\circ}$ plus $n\cdot 720^{\circ}$. –  Start wearing purple Oct 13 '13 at 13:59
Thanks again O.L. I finally understand it. So lets say if it's cos 5θ = 1/2, then I would need to 10 answers. –  Average_Ted Oct 13 '13 at 14:24
@Average_Ted Yes (if you are interested in the values between $0$ and $360$ degrees). –  Start wearing purple Oct 13 '13 at 14:47

No. $\cos 2\theta=\frac12$ implies that $2\theta = 60^\circ$ or $2\theta=300^\circ$ (not to forget $420^\circ$, $660^\circ$, $780^\circ$, $1020^\circ$ and so on). Therefore $\theta$ itself can be any of $30^\circ$, $150^\circ$, $120^\circ$, $330^\circ$ (and other values outside the $0^\circ\ldots 360^\circ$ range).

-

$$\cos\theta=\frac12=\cos60^\circ\implies \theta=n360^\circ\pm 60^\circ$$ where $n$ is any integer

$$\cos2\theta=\frac12=\cos60^\circ\implies 2\theta=m360^\circ\pm 60^\circ\implies \theta=m180^\circ\pm 30^\circ$$ where $m$ is any integer

In general, $\cos2\theta\ne 2\cos\theta$

-