Identities
Many identities interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity is written
where sin2 x + cos2 x is standard notation for (sin x)2 + (cos x)2.
Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.
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These in turn lead to the following three-angle formulae:
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
When three angles are equal, the three-angle formulae simplify to
These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations, much like the logarithm function.
Read more about this topic: Trigonometric Functions