## Trigonometric functions on the unit circle

*All* of the trigonometric functions of the angle *θ* can be constructed geometrically in terms of a unit circle centered at*O*.

The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (*x*, *y*) is a point of the unit circle, and if the ray from the origin (0, 0) to (*x*, *y*) makes an angle *t* from the positive *x*-axis, (where counterclockwise turning is positive), then

The equation *x*^{2} + *y*^{2} = 1 gives the relation

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

for any integer *k*.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(*x*_{1},*y*_{1}) on the unit circle such that an angle *t* with 0 < *t* < π/2 is formed with the positive arm of the *x*-axis. Now consider a point Q(*x*_{1},0) and line segments PQ OQ. The result is a right triangle ΔOPQ with ∠QOP = *t*. Because PQ has length *y*_{1}, OQ length *x*_{1}, and OA length 1, sin(*t*) = *y*_{1} and cos(*t*) = *x*_{1}. Having established these equivalences, take another radius OR from the origin to a point R(−*x*_{1},*y*_{1}) on the circle such that the same angle *t* is formed with the negative arm of the *x*-axis. Now consider a point S(*−x*_{1},0) and line segments RS OS. The result is a right triangle ΔORS 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 (−*x*_{1},*y*_{1}) is the same as (cos(π−*t*),sin(π−*t*)) and (*x*_{1},*y*_{1}) is the same as (cos(*t*),sin(*t*)), it is true that sin(*t*) = sin(π−*t*) and −cos(*t*) = cos(π−*t*). It may be inferred in a similar manner that tan(π−*t*) = −tan(*t*), since tan(*t*) = *y*_{1}/*x*_{1} and tan(π−*t*) = *y*_{1}/(−*x*_{1}). A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/sqrt(2).

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-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 like versine and exsecant – 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 calculated without the use of a calculator by using the Sum and Difference Formulas.

From: http://en.wikipedia.org/wiki/Unit_circle