cuquerob21

2023-03-01

How to find the exact value of $\frac{\mathrm{cos}\left(5\pi \right)}{6}$?

Centarxwm

Well, the odd multiples of the cosine $pI$ is always -1 hence $\mathrm{cos}\left(5\pi \right)=-1$ and
$\frac{\mathrm{cos}\left(5\pi \right)}{6}=-\frac{1}{6}$
Although if you mean $\mathrm{cos}\left(\frac{5\pi }{6}\right)$,
it can be found by using identity $\mathrm{cos}\left(\pi -x\right)=-\mathrm{cos}x$
and $\mathrm{cos}\left(\frac{5\pi }{6}\right)$
= $\mathrm{cos}\left(\pi -\frac{\pi }{6}\right)$
= $-\mathrm{cos}\left(\frac{\pi }{6}\right)$
= $-\frac{\sqrt{3}}{2}$

planelmolarvh8

There are two ways to complete the task, but if you remember the unit circle, you can complete them more quickly.
Convert $\frac{5\pi }{6}$ to angle degrees by using the equation:
$rad\cdot \left(\frac{180}{\pi }\right)=degrees$
$\frac{5\pi }{6}\cdot \frac{180}{\pi }={150}^{\circ }$
And we can figure out that the reference angle for ${150}^{\circ }$ is ${30}^{\circ }$.
${\mathrm{cos}30}^{\circ }=\frac{\sqrt{3}}{2}$
Since ${150}^{\circ }$ is in the 2nd quadrant, we know cosine is negative.
${\mathrm{cos}30}^{\circ }=\mathrm{cos}\left(\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2}$

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