Illustration of a unit circle. The variablet is anangle measure.Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. SinceC = 2πr, the circumference of a unit circle is2π.
If(x,y) is a point on the unit circle'scircumference, then|x| and|y| are the lengths of the legs of aright triangle whose hypotenuse has length 1. Thus, by thePythagorean theorem,x andy satisfy the equation
Sincex2 = (−x)2 for allx, and since the reflection of any point on the unit circle about thex- ory-axis is also on the unit circle, the above equation holds for all points(x,y) on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the openunit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", such as theRiemannian circle; see the article onmathematical norms for additional examples.
In thecomplex plane, numbers of unit magnitude are called theunit complex numbers. This is the set ofcomplex numbersz such that When broken into real and imaginary components this condition is
The complex unit circle can be parametrized by angle measure from the positive real axis using the complexexponential function, (SeeEuler's formula.)
Under the complex multiplication operation, the unit complex numbers form agroup called thecircle group, usually denoted Inquantum mechanics, a unit complex number is called aphase factor.
All of the trigonometric functions of the angleθ (theta) can be constructed geometrically in terms of a unit circle centered atO.Sine function on unit circle (top) and its graph (bottom)
Thetrigonometric functions cosine and sine of angleθ may be defined on the unit circle as follows: If(x,y) is a point on the unit circle, and if the ray from the origin(0, 0) to(x,y) makes anangleθ from the positivex-axis, (where counterclockwise turning is positive), then
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radiusOP from the originO to a pointP(x1,y1) on the unit circle such that an anglet with0 <t <π/2 is formed with the positive arm of thex-axis. Now consider a pointQ(x1,0) and line segmentsPQ ⊥ OQ. The result is a right triangle△OPQ with∠QOP =t. BecausePQ has lengthy1,OQ lengthx1, andOP has length 1 as a radius on the unit circle,sin(t) =y1 andcos(t) =x1. Having established these equivalences, take another radiusOR from the origin to a pointR(−x1,y1) on the circle such that the same anglet is formed with the negative arm of thex-axis. Now consider a pointS(−x1,0) and line segmentsRS ⊥ OS. The result is a right triangle△ORS with∠SOR =t. It can hence be seen that, because∠ROQ = π −t,R is at(cos(π −t), sin(π −t)) in the same way that P is at(cos(t), sin(t)). The conclusion is that, since(−x1,y1) is the same as(cos(π −t), sin(π −t)) and(x1,y1) is the same as(cos(t),sin(t)), it is true thatsin(t) = sin(π −t) and−cos(t) = cos(π −t). It may be inferred in a similar manner thattan(π −t) = −tan(t), sincetan(t) =y1/x1 andtan(π −t) =y1/−x1. A simple demonstration of the above can be seen in the equalitysin(π/4) = sin(3π/4) =1/√2.
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less thanπ/2. However, when defined with the unit circle, these functions produce meaningful values for anyreal-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions likeversine andexsecant – can be defined geometrically in terms of a unit circle, as shown at right.
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using theangle sum and difference formulas.
