Sine and Cosine Without a Calculator

Date: 9/2/96 at 1:27:21
From: Anonymous
Subject: Sine and Cosine Without a Calculator

How do you find the sine of 2pi/3 and the cosine of -2pi/3 
without using a calculator?

Date: 9/2/96 at 12:14:5
From: Doctor Ken
Subject: Re: Sine and Cosine Without a Calculator

Hello!

The easiest way to find Sines and Cosines is to use the picture 
method.  

   .

To draw one of these yourself, you would first make a unit circle, 
which is a circle of radius 1 with its center at (0,0).  Then you'd 
put your angle on the circle, with the vertex of your angle at (0,0) 
and one of the sides of the angle on the x-axis.  Then you measure 
positive angles COUNTER_CLOCKWISE, and negative angles clockwise.  
Your two angles, 2Pi/3 and -2Pi/3 , are already measured on the 
picture -- they're the long green lines.  The other two green lines in 
the picture are 3Pi/4 and 5Pi/6.  

To find the Sine and the Cosine of these angles, you look at the x- 
and y-coordinates of where these angles intersect the unit circle.  
The x-coordinate is the Cosine of the angle, and the y-coordinate is 
the Sine of the angle.  In the picture, I've labeled the main places 
on the axis where you'll need to know the measurements.

So, for instance, if your angle were 5Pi/6, then the y-coordinate 
of the intersection point would be 1/2 (that's the Sine) and the 
x-coordinate of the intersection point would be -Sqrt(3)/2 (that's 
the Cosine).  You can always check your work by using the formula 
Sin^2 + Cos^2 = 1.

Good luck!

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/