The sine, cosine and tangent are trigonometrical functions. Historically they arose as a ratio between the parties of a rectangular triangle therefore most convenient and to find them through a rectangular triangle. However through it it is possible to express only trigonometrical functions of acute angles. For obtuse angles it is necessary to enter a circle.

## It is required to you

- circle, rectangular triangle

## Instruction

1. Let in a rectangular triangle B corner - a straight line. AC will be a hypotenuse of this triangle, the party of AB and BC - its legs. A sine of an acute angle of BAC will be called the relation of BC leg, opposite to this corner, to AC hypotenuse. That is sin(BAC) = VS/AS.Kosinusom an acute angle of BAC will be called the relation of BC leg, adjacent to this corner, to AC hypotenuse. That is cos(BAC) = AB/AC. The cosine of the angle can also be expressed through a sine of the angle by means of the main trigonometrical identity: ((sin(ABC)) ^2)+ (cos (ABC)) ^2) = 1. Then cos(ABC) = sqrt (1 (sin (ABCs)) ^2). A tangent of an acute angle of BAC will be called the relation of BC leg, opposite to this corner, to AB leg, adjacent to this corner. That is tg(BAC) = BC/AB. The tangent of angle can also be expressed through its sine and a cosine on a formula: tg(BAC) = sin(BAC)/cos(BAC).

2. In rectangular triangles it is possible to consider only acute angles. For consideration of right angles it is necessary to enter a circle. Let O be the center of the Cartesian system of coordinates with axes X (an axis abtsiss) and Y (ordinate axis) and also the center of a circle of radius of R. The piece of OB will be the radius of this circle. Corners can be measured as turns from the positive direction of abscissa axis to OB beam. The direction counterclockwise is considered positive, clockwise negative. In designate a point abscissa for xB, ordinate - for uV.Togda the sine of the angle is defined as yB/R, a cosine of the angle - xB/R, a tangent of angle of tg (x) = by sin (x)/cos (x) = yB/xB.

3. The cosine of the angle can be calculated also in any triangle if lengths of all its parties are known. According to the theorem of cosines of AB^2 = ((AC) ^2)+ (BC) ^2) - 2*AC*BC*cos (ACB). From here, cos(ACB) = ((AC^2)+ (BC^2) (-AB^2)) / (2*AC*BC). The sine and a tangent of this corner can be calculated from definition of a tangent of angle and the main trigonometrical identity given above.