**Tangent** . Relative to an acute angle of a right triangle , ratio between the lengths of the opposite leg and the leg adjacent to the angle .

## Summary

[ hide ]

- 1 Reasons in the right triangle
- 2 Tangent
- 3 Tangent Values for Notable Angles of 30 °, 45 °, and 60 °
- 4 Sources

## Reasons in the right triangle

Figure 1

The ratios (ratios) between the lengths of the sides of a right triangle depend only on the amplitudes of their acute angles. Let’s take a closer look at this statement (Figure 1):

Let MAN be an acute angle.

From any point on one of its sides (B) other than vertex A, let us consider a line perpendicular to the other side, forming the triangle ABC rectangular in C, that is, with legs of lengths a and b, and hypotenuse c.

Let B ‘be any other point (B’ ≠ A) on the AM side and B be *any other point (B* ≠ A) on the AN side. Consider the perpendiculars B’C ‘and B *C* to AN and AM respectively. The three triangles ABC, AB’C ‘and AB *C*they have their equal angles (since they are rectangles and they have a common angle ), therefore they are similar, and as such their homologous sides are proportional.

## Tangent

These ratios between the lengths of the sides of a right triangle are of fundamental importance in the study of trigonometry . For an acute angle of the right triangle , the ratio between the length of the leg opposite the angle and the length of the leg adjacent to the angle is called the Tangent of the angle and is denoted by tan (tg), that is:

tan α = a / b, and tan β = b / a

Since α and β are complementary angles , in the previous relations it is observed that:

tan α = 1 / (tan (90 ° – α))

## Tangent values for notable 30 °, 45 °, and 60 ° angles

Figure 2

Consider a triangle ABC equilateral side 2 (figure 2). Let BD be perpendicular to B through AC. In triangle ADB right in D we have that the angle DAB measures 60 ° and the angle ABD measures 30 °, AB = 2, AD = 1, BD = √3, therefore:

tan 60 ° = √3

tan 30 ° = √3 / 3

To check how 45 ° = 1 it is enough to consider an isosceles right triangle .