It is easy to track behavior of trigonometrical functions, observing change of provision of a point on a single circle. And for fixing of terminology it is convenient to consider a ratio of the parties in a rectangular triangle.
To formulate definition of a tangent of angle and other trigonometrical functions, consider a ratio of corners and the parties in a rectangular triangle.
It is known that the sum of corners of any triangle is equal 180 °. Therefore, the sum of two indirect corners is equal in rectangular 90 °. The parties forming a right angle are called legs. The third party of a figure — a hypotenuse. Each of two acute angles of a rectangular triangle is formed by a hypotenuse and one leg which is called "adjacent" for this corner. Respectively, other leg is called "opposite".
Tanges of a corner is called the relation of an opposite leg to adjacent. It is in passing easy to remember that the return relation is called a corner cotangent. Then the tangent of one acute angle of a rectangular triangle is equal to a cotangent of the second. It is also obvious that the tangent of angle is equal to the relation of a sine of this corner to its cosine. The relation of the parties — the size which does not have dimension. The tangent as a sine, a cosine and a cotangent is a number. To each corner there corresponds the unique value of a tangent (sine, a cosine, a cotangent). Values of trigonometrical functions for any corner can be found in mathematical tables of Bradis. To learn what values the tangent of angle can accept, draw a single circle. At change of a corner from 0 ° to 90 ° the tangent changes from zero and directs in infinity. Function change nonlinear, on graphics is easy to find intermediate points for creation of a curve: tg 45 °=1, tg30 ° = 1 / √ 3, tg60 °= √ 3. For negative corners the tangent from zero directs in minus infinity. A tangent — periodic function with gaps at approach of value of an argument (corner) to 90 ° and -90 °.