By examining the unit circle, the following properties of the trigonometric functions can be established.
Symmetry
When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:
Note that the sign in front of the trig function does not necessarily indicate the sign of the value. For example, +cos θ does not always mean that cos θ is positive. In particular, if θ = π, then +cos θ = −1.
Shifts and periodicity
By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.
