Cosecant Cosecant, one of the six fundamental ratios of trigonometry.

Cosecant Cosecant, one of the six fundamental ratios of trigonometry. The others are sine, cosine, secant, tangent, and cotangent. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Cosecant embodies such a relationship between the magnitudes of the angles of a right triangle (a triangle with one 90° angle) and the lengths of the sides. Varying one value, such as the magnitude of an angle, requires the related value, such as the length of a side, to change in a predictable way. The cosecant, usually abbreviated csc, of one of the acute (less than 90°) angles of a right triangle is found by dividing the length of the hypotenuse (the triangle's longest side) by the length of the side opposite the angle: . Cosecant smoothly decreases in numerical value from infinity to 1 as the angle increases from 0° to 90°. Cosecant 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 cosecant of ? is equal to the length of the line connecting the point to the circle's center divided by the vertical distance of the point from the x axis. Cosecant smoothly increases in numerical value from 1 as ? increases from 90°, approaching infinity at 180°. The function is discontinuous at 180°; it breaks and begins rising from negative infinity to 1 as ? goes from 180° to 270°, then decreases again, approaching negative infinity as ? changes from 270° to 360°. Sine is cosecant's reciprocal function. The sine of an acute angle of a right triangle is equal to the length of the side opposite the chosen acute angle divided by the length of the hypotenuse: Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.