Unit Circle Sine Cosine

The Unit circle is used to understanding the sins and cos of angles to find 90 degree triangle. It is radius is exactly one. The center of circle is said to be origin and its perimeter comprises the set of all points that are exactly one unit from the center of the circle while placed in the plane.

It s just a circle with radius ‘one’.

The distance from the origin point(x,y) is `sqrt(X^2+Y^2)` by using Pythagorean Theorem.

Here radius is one So, The expression should becomes sqrt(x²+y²) =1

Take square on both sides then the eqn. becomes,

                      X²+y² =1

Positive angles are found using counterclockwise from the positive x axis

And negative angles are found using  anti clockwise from negative axis.

To find sine and cosine angle:

Angles are measures using counterclockwise from the positive x axis.The point where the ray at angle theta meets the unit circle is,
                                      (cos(`theta` ),Sin(`theta` )).
Here,cos(`theta` )----->Represents notation for the cosine of theta
Sine(`theta` ) is the notation of the sine of theta.
X= cos`(theta)`
Y=sin`(theta)`
The angle measured in radians is the length of the circular arc from the positive x
Axis counterclockwise until the desired angle is reached.
Since the radius of the unit circle is 1, the length of its perimeter is
2pr = 2p(1) = 2p.
Hence the angle for going exactly around the circle once is 2 radians. This
corresponds to 360 degrees.

Example problem -Unitcircle sines and cosine

Example 1: 

                 The angle q=0 radians is found by starting at the positive x-axis and going a distance 0. Hence the ray at angle q = 0 meets the unit circle at (1,0).

Hence
(cos(0), sin(0)) = (1,0),
cos(0) = 1,
sin (0)= 0.
Example 2. Without a calculator, find cos(p) and sin(p).
Solution:

The angle q= p radians is found by starting at the positive x-axis and going a distance p counterclockwise around the circle. But 2p is the total perimeter, so going a distance p arrives exactly half-way around the circle. Hence the ray at angle q = p meets the unit circle at (-1,0).

Hence
(cos(p), sin(p)) = (-1,0),
cos (p) = -1,
sin (p)= 0.