It is the relationship between the angles and the sides of a right-angle triangle.

Here according to the definition of trigonometry, we have to establish a relationship between the angle [math]\beta[/math] and the sides of the triangle.[br][br]So what we do is find the ratios of the sides of the triangle with respect to the angle [math]\beta[/math].[br]Here [b]CB[/b] is the side opposite to the angle commonly known as the [b]Opposite Side[/b].[br]And [b]BA[/b] is the side adjacent to the angle commonly known as the [b]Adjacent Side[/b].[br]And [b]CA[/b] is the [b]Hypotenuse[/b] irrespective of the angle.[br][br]So what the ancient mathematicians did is they gave unique names to the ratios of the sides of a right-angle triangle.[br][br]1. Sine[br][math]sin\beta=\frac{Opposite}{Hypotenuse}=\frac{CB}{CA}[/math][br][br]2. Cosine[br][math]cos\beta=\frac{Adjacent}{Hypotenuse}=\frac{AB}{CA}[/math][br][br]3. Tangent[br][math]tan\beta=\frac{Opposite}{Adjacent}=\frac{sin\beta}{cos\beta}=\frac{BC}{AB}[/math][br][br]These are the basic trigonometric ratios that were defined by our mathematicians.

One reason why mathematicians chose a right angle triangle is that value of ratios will remain a constant for a specific angle. [br][br]Let's see how it works:[br]Taking any two right-angle triangles with an angle equal there will be one common right-angle triangle and an angle common.[br]Using the AA similarity criterion of triangles we can say that all the right-angle triangles with some angle [math]\alpha[/math] will be similar.[br]And we also know that the ratio of sides of similar triangles will be the same because similar triangles are just scaled versions of each other.[br][br]Therefore, the sine of some angle [math]\alpha[/math] will remain a constant for any right-angle triangle you pick.[br]

Therefore this is how Ancient mathematicians defined a whole new way of measuring the sides of a triangle using angles.