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.

Unit-circle definitions

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used so far. The equation for the unit circle is:

Related Topics:
Unit circle - Circle

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: x^2 + y^2 = 1 ,

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In the picture, some common angles, measured in radians, are given. Measurements in the counter clockwise direction are positive angles and measurements in the clockwise direction are negative angles. Let a line making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.

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For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π:

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:sin heta = sinleft( heta + 2pi k ight)

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:cos heta = cosleft( heta + 2pi k ight)

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for any angle θ and any integer k.

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The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.

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Above, only sine and cosine were defined directly by the unit circle, but the other four trig functions can be defined by:

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: an heta = rac{sin heta}{cos heta} quad sec heta = rac{1}{cos heta}

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:csc heta = rac{1}{sin heta} quad cot heta = rac{cos heta}{sin heta}

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Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see below). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle). From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)

Related Topics:
India - Versin - Tangent - Secant line - Exsec

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