cuquerob21

2023-03-01

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

Centarxwm

Beginner2023-03-02Added 5 answers

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 $\mathrm{cos}(\pi -x)=-\mathrm{cos}x$

and $\mathrm{cos}\left(\frac{5\pi}{6}\right)$

= $\mathrm{cos}(\pi -\frac{\pi}{6})$

= $-\mathrm{cos}\left(\frac{\pi}{6}\right)$

= $-\frac{\sqrt{3}}{2}$

$\frac{\mathrm{cos}\left(5\pi \right)}{6}=-\frac{1}{6}$

Although if you mean $\mathrm{cos}\left(\frac{5\pi}{6}\right)$,

it can be found by using identity $\mathrm{cos}(\pi -x)=-\mathrm{cos}x$

and $\mathrm{cos}\left(\frac{5\pi}{6}\right)$

= $\mathrm{cos}(\pi -\frac{\pi}{6})$

= $-\mathrm{cos}\left(\frac{\pi}{6}\right)$

= $-\frac{\sqrt{3}}{2}$

planelmolarvh8

Beginner2023-03-03Added 4 answers

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.

$\mathrm{cos}30}^{\circ}=\mathrm{cos}\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2$

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.

$\mathrm{cos}30}^{\circ}=\mathrm{cos}\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2$