Sine. Sine, one of the fundamental ratios of trigonometry. A ratio

Sine. Sine, one of the fundamental ratios of trigonometry. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Sine embodies the relationship between the lengths of the sides of a right triangle (a triangle with a 90° angle) and the magnitudes of its angles. This relationship means that varying one value, such as the length of a side, requires another value, such as the magnitude of an angle, to change in a predictable way. The sine, usually abbreviated sin, of one of the acute (less than 90°) angles of a right triangle is equal to the length of the side opposite the acute angle divided by the length of the longest side, called the hypotenuse: . Sine smoothly increases in numerical value from 0 to 1 as the angle increases from 0° to 90°. Sine is also defined for angles greater than 90° using right triangles inscribed in a circle centered at the point (0,0) on the xy axis: A line drawn from the circle's center to any point on the circle makes an angle, ? , with the x axis. The sine of ? is equal to the vertical distance of the point from the x axis divided by the length of the line connecting the point to the circle's center. Sine smoothly decreases in numerical value from 1 to -1 as ? increases from 90° to 270° and then increases again to 0 as ? goes from 270° to 360°. Cosecant is sine's reciprocal function. The cosecant, usually abbreviated csc, of an acute angle of a right triangle is equal to the length of the triangle's hypotenuse divided by the length of the side opposite the chosen acute angle: . See also Cosine, Secant, Tangent, Cotangent. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.