How's about this................sine of zero degrees is zero..............the sine of 30 degrees is 1/2...........the sine of 45 degrees is 1/2 the square root of 2............the sine of 60 degrees is 1/2 the square root of 3................the sine of 90 degrees is 1...............................0 and 1..................right between them is 1/2............................the full square root of 2 added to the full square root of 3........is very close to pi.........................relatively theory................Dr. Einstein..............

Unit Circle

The "Unit Circle" is a circle with a radius of 1.

Being so simple, it is a great way to learn and talk about lengths and angles.
The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.

Sine, Cosine and Tangent

Because the radius is 1, we can directly measure sine, cosine and tangent.

What happens when the angle, θ, is 0°?

cos 0° = 1, sin 0° = 0 and tan 0° = 0

What happens when θ is 90°?

cos 90° = 0, sin 90° = 1 and tan 90° is undefined

Try It Yourself!

Have a try! Move the mouse around to see how different angles (in radians or degrees) affect sine, cosine and tangent

The "sides" can be positive or negative according to the rules of Cartesian coordinates. This makes the sine, cosine and tangent change between positive and negative values also.

Also try the Interactive Unit Circle.

Pythagoras

Pythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:

x² + y² = 1²

But 1² is just 1, so:

x² + y² = 1
(the equation of the unit circle)

Also, since x=cos and y=sin, we get:

(cos(θ))² + (sin(θ))² = 1

a useful "identity"