Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?
hehehehe... while pasting forgot to change that pi/3
but if you draw your self that angle ... you'll realize that angle of is the same as the angle of and you know that angle that is
because cos is positive in first and 4th quadrant get it ?
when drawn circle, draw and add it 5 times and you'll see that is the angle
P.S. you realize that so is just for the smaller than so it have to be
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...
That's
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