Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation

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:y,=-y

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i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. See the trigonometric identity article for this technique.

Related Topics:
Vector space - Basis - Linear differential equation - Trigonometric identities

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The tangent function is the unique solution of the nonlinear differential equation

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:y,'=1+y^2

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satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation; see http://www.usfca.edu/vca/PDF/vca-preface.pdf.

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The significance of radians

Radians constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

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f(x) = sin(kx); k e 0, k e 1 ,

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then the derivatives will scale by amplitude:

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f'(x) = kcos(kx) ,.

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k here is a constant that represents a mapping between units. If x is in degrees, k = rac{pi}{180}.

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This means that the second derivative of a sine in degrees satisfies not the differential equation

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y = -y ,,

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but

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y = -k^2y ,;

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similarly for cosine.

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This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

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