The ratios of the lengths of different sides of a right angled triangle are known as trigonometric ratios. These ratios are sine, cosine, tangent, cotangent, cosecant, secant. The sides of a right angle triangle can be seen as given here.

In a right angled triangle the tangent of an angle is the ratio of the length of the perpendicular to the ratio of the length of its base. The perpendicular can be termed as adjacent and the base as opposite.

The tangent of an angle can be measured in degree or radians. The opposite of tangent is known as cotangent. The cotangent of an angle is the ratio of the length of the base to the length of the perpendicular.

## Definition

The tangent of an angle can be defined for a right angled triangle as the ratio of the length of the perpendicular to the ratio of the length of its base. The tangent ratio of a triangle will remain same irrespective of the change in the size of the triangle.  In the given figure, tan A will be the ratio of opposite to adjacent, that is, the ratio of perpendicular to the base.

The tangent has one more definition with respect to a curve. It can be defined as the line touching a curve at a single point. The given figure will show the relation between tangent of an angle and tangent to a curve.

For the given unit circle, the tangent of angle OAB is the length of the segment AB.

## How to find tangent of an angle

To find the tangent of an angle we need to know the length of base and perpendicular. If it is not given, we need to obtain them and then take their ratio to find the tangent of the angle. For example, we need to find the tangent of angle for a right angled triangle with base 4 units, and hypotenuse 5 units. To solve this, first we will find the perpendicular of the triangle using the Pythagoras theorem.
Perpendicular = $\sqrt{(5^{2}-4^{2})} = \sqrt{25-16} = 3$
The value of tangent of angle = $\frac{3}{4}$

## Sine and Cosine of an Angle

There are two major trigonometric ratios, apart from tangent, that are obtained from a right angled triangle. They are sine and cosine. Sine of an angle is the ratio of the length of the perpendicular to the length of hypotenuse. Cosine is the ratio of the length of the base to the length of the hypotenuse. Cosecant is the reciprocal of sine and secant is the reciprocal of cosine.

 $sin\theta$ $\frac{p}{h}$ $cos\theta$ $\frac{b}{h}$ $tan\theta$ $\frac{p}{b}$

## Examples

Example 1: For a right angled triangle, the length of perpendicular is given to be 8 and the length of the hypotenuse as 17. Find the tangent of the angle A for the given triangle.

Solution: The length of the base can be calculated by using the Pythagoras theorem. Now we have,

Base = $\sqrt{(17^{2}-8^{2})} = \sqrt{289-64} = 15$

To calculate the tangent of angle we need the ratio of length of the perpendicular to that of the base.

tan A = $\frac{8}{15}$

Example 2: Find the value of tan A if sin A = $\frac{3}{8}$ in the given triangle.

Solution: We have, sin A = $\frac{3}{8}$ = $\frac{p}{h}$ where p is the perpendicular and b is the base of the triangle. Now, we need to know the length of the base, b, so that we can calculate tan A. Using the pythagoras theorem we have,

Base = $\sqrt{(8^{2}-3^{2})} = \sqrt{64-9} = \sqrt{55}$

Hence, tan A = $\frac{3}{\sqrt{55}}$.