Do you have any techniques to know sin and cosine using radians without using a caculator?

Question

I just wanted to have a technique so that I will not be confused. e.g.: 16456pi/6 and 78pi/3, for pi/3, pi/6, and pi/4 only. Thanks.

Answer 1

16456pi/6 = 2742pi + 2pi/3 = 1371(2pi) + 2pi/3
The first part you can simply throw out because sin and cos are periodic functions - so that sin(x+2pi) = sin(x). Even if you add the (2pi) 1371 times, the value won't change.
Now remember that sin(x) = sin(pi-x), and cos(x)=-cos(pi-x)
so sin(2pi/3) = sin(pi-2pi/3) = sin (pi/3) and
cos(2pi/3) = -cos(pi-2pi/3) = -cos (pi/3)

Same approach for 78pi/3 - first find out how many (2pi)s in there, and get rid of it: 78pi/3 = 26pi = 13*(2pi) + 0
So, you are left with just 0. sin(0) = 0; cos(0)=1. That's it.

Answer 2

using the property sin(2pi+x) = sin x
reduce the angle to range 0 to 2pi

use standard values for 0, pi/6, pi/4, pi/3 etc for the range of angles 0 to pi/2
for pi/2 to pi
sin x = -sin(pi-x)
for pi to 3pi/2
sin x = -sin(x-pi)
for 3pi/2 to 2pi
sinx = sin(2pi-x)
for pi to 2pi use similar formula
for other values table look up or series expansion'

Answer 3

Use the unit circle. Divide all those radian measures and what ever's the remainder should be found with it.

Answer 4

The angles you mention all have exact solutions based on the pythagorean theorem. You can read about them at http://en.wikipedia.org/wiki/Special_right_triangles . You can read about methods for exact calculations of an even wider variety of angles at http://en.wikipedia.org/wiki/Exact_trigonometric_constants .

Answer 5

find the remainder from mod 360