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The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Similar right triangles illustrating the tangent and secant trigonometric functions Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
De Moivre's Theorem for Trig Identities by Michael Croucher, Wolfram Demonstrations Project Listen to this article ( 18 minutes ) This audio file was created from a revision of this article dated 5 June 2021 ( 2021-06-05 ) , and does not reflect subsequent edits.
The proof is valid for almost all right triangles, except for the case when a=b, i.e. Isosceles Right Triangle. For the case when a ≠ b {\displaystyle a\neq b} , we can always choose to orient the right triangle A B C {\displaystyle \triangle ABC} so that a<b as shown in the figure.