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

• Sep 15th 2010, 10:17 AM
jayshizwiz
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?
• Sep 15th 2010, 10:36 AM
yeKciM
Quote:

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 :D

P.S. you should know by heart this :D

$\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 :D:D:D:D
• Sep 15th 2010, 10:53 AM
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?
• Sep 15th 2010, 11:00 AM
yeKciM
Quote:

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 :D:D:D

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 :D:D:D:D get it ?

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

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 $-\frac {\pi}{3}$
• Sep 15th 2010, 11:09 AM
undefined
Quote:

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.
• Sep 15th 2010, 02:15 PM
ebaines
Quote:

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?

The key concept her is to realize that 1 radian is the angle that is covered if you wrap a string of length 1 x radius around the perimeter of a circle. By convention we usually just consider a circle of radius 1, which would have a total perimeter length 2 pi. Hence a string of length 2 pi would wrap precisely once around this unit circle, and so the angle 2 pi radians is eqivalent to 360 degrees. Similarly, pi/2 radians is 180 degreees, pi/3 radians = 120 degrees, etc. Now, when first learning trig there is a natural tendency for some students to resist thinking in radians, and try to convert everything to degrees in their head. But trust me - you'd be much beter off getting to a point where you can "visualize" what pi/4 radians means without first thinking about degrees. Just consider that each quadrant of the circle is pi/2 radians, and so 2/3 pi is in the second quadrant, 5/3 pi is in the fourth, etc. So get used to visualizing how it is that tan(pi/4) = 1, and you'll be well on your way.
• Sep 15th 2010, 05:19 PM
Quote:

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 $Cos60^o=0.5$