Trigonometric functions

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). For angles less than a right angle, trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the arguments to the whole number line and to the complex numbers.

Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.

Right-angled triangle definitions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.
Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants.
Bottom: Graph of sine function versus angle. Angles from the top panel are identified.
Plot of the six trigonometric functions and the unit circle for an angle of 0.7 radians.

The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

  • The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
  • The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
  • The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° (π radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of (0, π/2) as expressed in interval notation. The following definitions apply to angles in this (0, π/2) range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin(θ) for angles θ, πθ, π + θ, and 2πθ depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.

Function Abbreviation Description Identities (using radians)
sine sin opposite/hypotenuse
cosine cos adjacent/hypotenuse
tangent tan (or tg) opposite/adjacent
cotangent cot (or cotan or cotg or ctg or ctn) adjacent/opposite
secant sec hypotenuse/adjacent
cosecant csc (or cosec) hypotenuse/opposite

Sine, cosine, and tangent

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus for gulf or bay,[3] since, given a unit circle, it is the side of the triangle on which the angle opens. In our case:

An illustration of the relationship between sine and its out-of-phase complement, cosine. Cosine is identical, but π/2 radians out of phase to the left; so cos A = sin(A + π/2).

The cosine (sine complement, Latin: cosinus, sinus complementi) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle.[4] Because the angle sum of a triangle is π radians, the co-angle B is equal to π/2A; so cos A = sin B = sin(π/2A). In our case:

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch).[5] In our case:

Tangent may also be represented in terms of sine and cosine. That is:

These ratios do not depend on the size of the particular right triangle chosen, as long as the focus angle is equal, since all such triangles are similar.

The acronyms "SOH-CAH-TOA" ("soak-a-toe", "sock-a-toa", "so-kah-toa") and "OHSAHCOAT" are commonly used trigonometric mnemonics for these ratios.

Secant, cosecant, and cotangent

The remaining three functions are best defined using the three functions above and can be considered their reciprocals.

The secant of an angle is the reciprocal of its cosine, that is, the ratio of the length of the hypotenuse to the length of the adjacent side, so called because it represents the secant line that cuts the circle (from Latin: secare, to cut):[6]

The cosecant (secant complement, Latin: cosecans, secans complementi) of an angle is the reciprocal of its sine, that is, the ratio of the length of the hypotenuse to the length of the opposite side, so called because it is the secant of the complementary or co-angle:

The cotangent (tangent complement, Latin: cotangens, tangens complementi) of an angle is the reciprocal of its tangent, that is, the ratio of the length of the adjacent side to the length of the opposite side, so called because it is the tangent of the complementary or co-angle:

Mnemonics

Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the rise, run, and slope of a line segment relative to horizontal. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine and tangent. With a line segment length of 1 (as in a unit circle), the following mnemonic devices show the correspondence of definitions:

  1. "Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise when the length of the line is 1.
  2. "Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run when the length of the line is 1.
  3. "Tangent combines the rise and run" meaning that Tangent takes the angle of the line segment and tells its slope, or alternatively, tells the vertical rise when the line segment's horizontal run is 1.

This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e. angles and slopes. (The arctangent or "inverse tangent" is not to be confused with the cotangent, which is cosine divided by sine.)

While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have a length of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the run at an angle of 7° is 5cos(7°).

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