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 \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
hehehehe... while pasting forgot to change that pi/3
but if you draw your self that angle ... you'll realize that angle of $\displaystyle \displaystyle \frac {5\pi}{3} $ is the same as the angle of $\displaystyle \displaystyle -\frac{\pi}{3} $ and you know that angle that is $\displaystyle \displaystyle \cos {-\frac {\pi}{3} } = \frac {1}{2} $
because cos is positive in first and 4th quadrant get it ?
when drawn circle, draw $\displaystyle \frac {\pi}{3} $ and add it 5 times and you'll see that is the angle $\displaystyle -\frac {\pi}{3} $
P.S. you realize that $\displaystyle \displaystyle \frac {6\pi}{3} == 2\pi $ so $\displaystyle \displaystyle \frac {5\pi}{3} $ is just for the $\displaystyle \frac {\pi}{3} $ smaller than $\displaystyle 2\pi$ so it have to be $\displaystyle -\frac {\pi}{3} $
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.
Without using a calculator...
$\displaystyle \displaystyle\frac{5{\pi}}{3}=\frac{5(180^o)}{3}=3 00^o$
That's $\displaystyle -60^o$
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 $\displaystyle Cos60^o=0.5$