Trigonometry is an important branch of Mathematics. It provides the relationships between the lengths and angles of a triangle, especially the right-angled triangle. Also, it will cover many other geometrical shapes like circles. There are three main trigonometry ratios such as sine, cosine, and tangent. Tangent is one of these and popularly known as ‘tan’ of some angle. This topic will explain the tangent formula with examples. Let us learn it!

**What is Trigonometry?**

It is the study of the relationships which involves angles, lengths, and heights of triangles given. It also relates to the different parts of circles as well as other geometrical figures. Trigonometry has many trigonometric ratios which are very fundamental in mathematics. It has many identities that are very useful for learning and deriving the many equations and formulas in science. There are many fields where these identities of trigonometry and formula of trigonometry are useful.

**What is the Tangent Function?**

Tangent is the ratio of the opposite side divided by the adjacent side in a right-angled triangle. In trigonometry, there are six possible ratios. A ratio is a comparison of two numbers i.e. sides of a triangle. The Greek letter, \(\theta\), will be used to represent the reference angle in the right triangle. These six ratios are useful in different ways to compare two sides of a right triangle.

Tangent Angle Formula is normally useful to calculate the angle of the right triangle. In a right triangle, the tangent of an angle is a simple ratio of the length of the opposite side and the length of the adjacent side. Tangent is usually denoted as ‘tan’, but it is pronounced as a tangent. This function is useful to find out the length of a side of a triangle. It is possible when someone knows at least one side of the triangle and one of the acute angles.

The tangent function, along with sine and cosine functions, is one of the three most common trigonometric functions. Also in trigonometry, we may represent tan \(\theta\) as the ratio of sin \(\theta\) and cos \(\theta.\)

**Formula for a Tangent**

We will consider the right-angled triangle. In these, we have three sides namely – Hypotenuse, the opposite side (Perpendicular) and Adjacent side (Height). The largest side is the hypotenuse, the side opposite to the angle is opposite and the side where both hypotenuse and opposite rests is the adjacent side.

In a right-angled- triangle,

\(Tan\theta = \frac{{perpendicular}}{{base}} = \frac{y}{x} = \frac{{Sin\theta }}{{Cos\theta }}\)

The trigonometry ratios are having many real-world and practical applications in fields like aviation, architecture, surveying. Using the trigonometric ratios, such as tangent, allows for the measurement of things that cannot be determined using typical measurement tools. Height and distance problems are using the tangent formula.

**Solved Examples for ****Tangent Formula**

Q.1: Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 8 cm and 6 cm respectively?

Solution: Given,

Adjacent side i.e. base = 8 cm

Opposite side i.e. perpendicular = 6 cm

Also, the tangent formula is:

\(Tan\theta = \frac{{perpendicular}}{{base}}\)

i.e.\( tan\theta = \frac {6} {8}\)

\(tan \theta = 0.75\)

Thus tangent value will be 0.75.