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In mathematics, the trigonometric functions (also called the circular functions) 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 originating at the origin and making some angle with the ''x''-axis, the sine of the angle gives the length of the ''y''-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of 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). More precise definitions are detailed below. Trigonometric functions 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 arbitrary positive and negative values and even to 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== 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°,90°) as expressed in interval notation. The following definitions apply to angles in this 0° – 90° 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'', ''π'' − ''θ'', ''π'' + ''θ'', and 2''π'' − ''θ'' 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 2''π''. 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. - \theta \right) = \frac |- ! cosine | cos |align=center|adjacent / hypotenuse | |- ! tangent | tan (or tg) |align=center|opposite / adjacent | |- ! cotangent | cot (or cotan or cotg or ctg or ctn) |align=center|adjacent / opposite | |- ! secant | sec |align=center|hypotenuse / adjacent | |- ! cosecant | csc (or cosec) |align=center|hypotenuse / opposite | |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Trigonometric functions」の詳細全文を読む スポンサード リンク
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