The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
The algebraic expressions for the most important angles are as follows:
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
Sine and cosine can be defined as the unique solution to the initial value problem:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is of great importance in complex analysis:
For the proof of this expansion, see Sine. From this, it can be deduced that
This formula is commonly considered for real values of x, but it remains true for all complex values.
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
One can also define the trigonometric functions using various functional equations.
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.