# How to find $\arccos(\cos15\pi/11)$?

How to find $\arccos(\cos15\pi/11)$?

I'm completely lost. $15\pi/11$ isn't on the unit circle so how do I find the cosine of it?

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Do you understand radians? Pi radians are 180 degrees, 2pi radian to a full circle. – JoeTaxpayer Dec 6 '13 at 18:14
@JoeTaxpayer I mean that 15pi/11 isn't on the unit circle in the sense that it's not one of the generic ones like pi/3 or 3pi/2. So how do I find the cosine of something that's not one of the generic values on the unit circle? – Anon Dec 6 '13 at 18:21
Sorry. I understand now. Using a calculator isn't an option I suppose? A large enough well drawn circle will let you interpolate interim values. 1.3636 pi. – JoeTaxpayer Dec 6 '13 at 18:24

As $\cos x=\cos A\implies x=2n\pi\pm A$ where $n$ is any integer

The general value of $$\arccos\left(\cos\frac{15\pi}{11}\right)=2n\pi\pm \frac{15\pi}{11}$$

Based on the definition of principal value inverse cosine ratio, $\displaystyle0\le \arccos x\le\pi$

Taking '+' sign, $\displaystyle0\le 2n\pi+\frac{15\pi}{11}\le\pi\implies 0\le 22n+15\le 11\implies0\le n<0$ which is impossible

Taking '-' sign, $\displaystyle0\le 2n\pi-\frac{15\pi}{11}\le\pi\implies 0\le 22n-15\le 11\implies 1\le n\le1\implies n=1$

So, the principal value is $\displaystyle 2\pi-\frac{15\pi}{11}=??$

Alternatively, as we know $\cos\left(\pi\pm y\right)=-\cos y$

$$\cos\left(\frac{15\pi}{11}\right)=\cos\left(\pi+\frac{4\pi}{11}\right)=-\cos\left(\frac{4\pi}{11}\right)=\cos\left(\pi-\frac{4\pi}{11}\right)=\cos\left(\frac{7\pi}{11}\right)$$

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"15pi/11 isn't on the unit circle" - Do you think OP will find 7pi/11 any easier? – JoeTaxpayer Dec 6 '13 at 18:16
@JoeTaxpayer, does not matter. I am supplying a generic solution – lab bhattacharjee Dec 6 '13 at 18:18
@labbhattacharjee I don't quite understand your solution. I understand radians. – Anon Dec 6 '13 at 18:20
@Anon, please find the first method – lab bhattacharjee Dec 6 '13 at 18:27
@JoeTaxpayer, would you mind checking the edited answer? – lab bhattacharjee Dec 6 '13 at 18:30