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,) |codomain=() |period=2''π'' | zero=0 |plusinf= |minusinf= |max=((2''k'' + ½)''π'', 1) |min=((2''k'' − ½)''π'', −1) | vr1= |f1= |vr5= |f5= | asymptote= |root=''kπ'' |critical=''kπ'' − π/2 |inflection=''kπ'' |fixed=0 | notes = }} Sine, in mathematics, is a trigonometric function of an angle. The sine of an angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (i.e., the hypotenuse). 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. The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the ''jyā'' and ''koṭi-jyā '' functions used in Gupta period Indian astronomy (''Aryabhatiya'', ''Surya Siddhanta''), via translation from Sanskrit to Arabic and then from Arabic to Latin.〔Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7, p. 210.〕 The word "sine" comes from a Latin mistranslation of the Arabic ''jiba'', which is a transliteration of the Sanskrit word for half the chord, ''jya-ardha''.〔() Victor J Katz, A history of mathematics, p210, sidebar 6.1.〕 == Right-angled triangle definition == For any similar triangle the ratio of the length of the sides remains the same. For example, if the hypotenuse is twice as long, so are the other sides. Therefore respective trigonometric functions, depending only on the size of the angle, express those ratios: between the hypotenuse and the "opposite" side to an angle ''A'' in question (see illustration) in the case of sine function; or between the hypotenuse and the "adjacent" side (cosine) or between the "opposite" and the "adjacent" side (tangent), etc. To define the trigonometric functions for an acute angle ''A'', start with any right triangle that contains the angle ''A''. The three sides of the triangle are named as follows: * The ''adjacent side'' is the side that is in contact with (adjacent to) both the angle we are interested in (angle ''A'') and the right angle, in this case side b. * 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. 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 greater than 0° and less than 90°. The following definition applies to such angles. The angle ''A'' (having measure α) is the angle between the hypotenuse and the adjacent side. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case, it does not depend on the size of the particular right triangle chosen, as long as it contains the angle ''A'', since all such triangles are similar. 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「sine」の詳細全文を読む スポンサード リンク
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