# Thread: How do you convert from cos(pi/number)

1. ## How do you convert from cos(pi/number)

Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?

2. Originally Posted by jayshizwiz
Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?

draw your self circle ... with that angle ... and you'll see

P.S. you should know by heart this

$\displaystyle
\begin{matrix}
\alpha & 0° & \frac {\pi}{6} == 30° & \frac {\pi}{4} == 45° & \frac {\pi}{3} == 60° & \frac {\pi}{2} == 90° & \pi == 180° & \frac {3\cdot \pi}{2} == 270° &2\pi == 360° \\
\sin{\alpha} &0 & \frac {1}{2} & \frac {\sqrt{2}}{2} & \frac {\sqrt{3}}{2} & 1 &0 & -1& 0\\
\cos{\alpha} & 1 & \frac {\sqrt{3}}{2} & \frac {\sqrt{2}}{2} & \frac {1}{2} & 0 &-1 &0 &1 \\
\tan{\alpha}& 0& \frac {\sqrt{3}}{3} & 1 & \sqrt{3} & \pm \infty & 0 & \mp \infty & 0\\
\csc{\alpha} & \pm \infty & \sqrt{3} & 1 & \frac {\sqrt{3}}{3} & 0 & \mp \infty &0 & \pm \infty
\end{matrix}
$

sorry for bad table ... lol forgot how to draw table

3. thanks but hopefully someone has a better explanation (:

and you wrote pi/6 twice...

I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

but isn't there a way to measure it by radian?

4. Originally Posted by jayshizwiz
thanks but hopefully someone has a better explanation (:

and you wrote pi/6 twice...

I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

but isn't there a way to measure it by radian?
hehehehe... while pasting forgot to change that pi/3

but if you draw your self that angle ... you'll realize that angle of $\displaystyle \frac {5\pi}{3}$ is the same as the angle of $\displaystyle -\frac{\pi}{3}$ and you know that angle that is $\displaystyle \cos {-\frac {\pi}{3} } = \frac {1}{2}$
because cos is positive in first and 4th quadrant get it ?

when drawn circle, draw $\frac {\pi}{3}$ and add it 5 times and you'll see that is the angle $-\frac {\pi}{3}$

P.S. you realize that $\displaystyle \frac {6\pi}{3} == 2\pi$ so $\displaystyle \frac {5\pi}{3}$ is just for the $\frac {\pi}{3}$ smaller than $2\pi$ so it have to be $-\frac {\pi}{3}$

5. Originally Posted by jayshizwiz
Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?
yeKciM's answer is good, also you can use identities, for example cos(5pi/3) = cos(-5pi/3) = cos(-5pi/3 + 2pi) = cos(pi/3), then you can draw a 30-60-90 triangle if you don't have it memorised.

6. Originally Posted by jayshizwiz
I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

but isn't there a way to measure it by radian?

7. Originally Posted by jayshizwiz
Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator ?
Without using a calculator...

$\displaystyle\frac{5{\pi}}{3}=\frac{5(180^o)}{3}=3 00^o$

That's $-60^o$

Hence, if you draw a unit-circle centred at the origin, you can draw a right-angled triangle
with hypotenuse going off at 60 degrees to the x-axis,
since Cos(angle) gives the x co-ordinate, and Cos(-A)=CosA.

This allows us to draw a regular hexagon, made of equilateral triangles as shown in the attachment.

From that, we can see that $Cos60^o=0.5$