Trigonometric ratios

Introduction

Trigonometric ratios of an acute angle in the right angle triangle express the relationship between the angle and length of its sides.

Fig. 1

Definition

Consider the right angle triangle ABC shown in Fig. 1. Here the angle A (Angle CAB) is an acute angle. Side BC is facing the angle A. Hence BC is the side opposite to angle A. AC is the hypotenuse of the the right angle triangle ABC and the side AB is part of the angle A. Hence AB is the side adjacent to angle A.

Trigonometric ratios of the angle A in the right angle triangle ABC are defined as follows.$sine\ of\ the\ angleA=sin\ A={\frac {side\ opposite\ to\ angle\ A}{hypotenuse}}={\frac {BC}{AC}}$

$cosine\ of\ the\ angleA=cos\ A={\frac {side\ adjacent\ to\ angle\ A}{hypotenuse}}={\frac {AB}{AC}}$

$tangent\ of\ the\ angleA=tan\ A={\frac {side\ opposite\ to\ angle\ A}{side\ adjacent\ to\ angle\ A}}={\frac {BC}{AB}}$

$cosecant\ of\ the\ angleA=cosec\ A={\frac {1}{sin\ A}}={\frac {hypotenuse}{side\ opposite\ to\ angle\ A}}={\frac {AC}{BC}}$

$secant\ of\ the\ angleA=sec\ A={\frac {1}{cos\ A}}={\frac {hypotenuse}{side\ adjacent\ to\ angle\ A}}={\frac {AC}{AB}}$

$cotangent\ of\ the\ angleA=cot\ A={\frac {1}{tan\ A}}={\frac {side\ adjacent\ to\ angle\ A}{side\ opposite\ to\ angle\ A}}={\frac {AB}{BC}}$

$tan\ A={\frac {BC}{AB}}={\frac {\frac {BC}{AC}}{\frac {AB}{AC}}}={\frac {sin\ A}{cos\ A}}$

$cot\ A={\frac {AB}{BC}}={\frac {\frac {AB}{AC}}{\frac {BC}{AC}}}={\frac {cos\ A}{sin\ A}}$

Note: Here the symbol sin A is used as an abbreviation for ''the sine of the angle A". sin A is not the product of "sin" and A. "sin" separated from A. Similar interpretation follow for other trigonometric ratios.

The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.