Trigonometry

Trigonometry is the study of angles and of the angular relationships of planar and three-dimensional figures. The trigonometric functions (also called circular functions) are the cosecant , cosine , cotangent , secant , sine , and tangent . Their inverse trigonometric functions, with appropriate branch conventions, are denoted , , , , , and . Note that the notation here means inverse function, not to the power.

The trigonometric functions are most simply defined using the unit circle. Let be an angle in radians measured counterclockwise from the x-axis along an arc of the circle. Then is the horizontal coordinate of the arc endpoint, and is the vertical component. The ratio is defined as . As a result of this definition, the trigonometric functions are periodic, all satisfying periodicity wherever they are defined, so

(1)

where is an integer and func is a trigonometric function. The tangent and cotangent also have period .

Replacing the unit circle by a superellipse leads to the generalized subject known as squigonometry.

A right triangle has three sides: the hypotenuse and, relative to a given acute angle , the adjacent side and opposite side. A helpful mnemonic for remembering the definitions of the trigonometric functions is then given by "oh, ah, o-a," "Soh, Cah, Toa," or "SOHCAHTOA", i.e., sine equals the opposite side over the hypotenuse, cosine equals the adjacent side over the hypotenuse, and tangent equals the opposite side over the adjacent side,

			(2)
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Another mnemonic probably more common in Great Britain than the United States is "Tommy On A Ship Of His Caught A Herring."

The Pythagorean identity, which follows from the Pythagorean theorem, is

(5)

It is therefore also true that

(6)

and

(7)

The trigonometric functions can be defined algebraically in terms of complex exponentials (i.e., using the Euler formula) as

			(8)
			(9)
			(10)
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Hybrid trigonometric product/sum formulas are

			(19)
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			(22)

Osborn's rule gives a prescription for converting trigonometric identities to analogous identities for hyperbolic functions.

For imaginary arguments,

			(23)
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For complex arguments,

			(25)
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For the absolute square of a complex number with ,

			(27)
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The complex modulus also satisfies the curious identity

(29)

For an entire function , the identity

(30)

holds for all real and exactly when , , or , with and (Robinson 1957; Haruki 1966, p. 473).