How to determine cosine

Cosine is the basic trigonometric function of the angle. The ability to determine cosine is useful in vector algebra when defining the projections of vectors on different axes.

Instruction manual

one

The cosine of an angle is the relation of the adjacent to the corner of the leg to hypotenuse. Therefore, in a right-angled triangle ABC (ABC is a right angle), the cosine of the angle BAC is equal to the ratio of AB to AC. For angle ACB: cos ACB = BC / AC.

2

But not always the angle belongs to the triangle, in addition there are obtuse angles that obviously cannot be part of a right-angled triangle. Consider the case when the angle is given by rays. In order to calculate the cosine of the angle in this case, proceed as follows. A coordinate system is tied to the corner, the origin is calculated from the top of the corner, the X axis goes along one side of the corner, the Y axis is built perpendicular to the X axis. Then, a circle of unit radius centered at the top of the corner is built. The second side of the angle intersects the circle at point A. Lower the perpendicular from point A to the X axis, mark the point of intersection of the perpendicular with the axis Ax. Then we get a right triangle AAxO, and the cosine of the angle is AAx / AO. Since the circle is of unit radius, then AO = 1 and the cosine of the angle is simply AAx.

3

In the case of an obtuse angle, all the same constructions are carried out. The oblique angle cosine is negative, but it is also equal to Ax.

note

The cosines of some angles are presented in the Bradis tables.