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.

It is a circle with a radius of 1 unit. This is very useful for calculating trigonometric ratios.[br]Here, [br]O is the center.[br]A is any point on the circumference of the circle.[br]B is the perpendicular from A to the x-axis.[br]E is the x-intercept of the unit circle.[br]C is the x-intercept of the tangent drawn to the circle at A.[br]D is the y-intercept of the tangent drawn to the circle at A.

[math]sin\Theta=\frac{opp}{hyp}=\frac{AB}{OA}=\frac{AB}{1}=AB[/math][br]Therefore sin theta is equal to the length of AB or the value of the y coordinate of the point A.

[math]cos\Theta=\frac{adj}{hyp}=\frac{OB}{OA}=\frac{OB}{1}=OB[/math][br]Therefore cos theta is equal to the length of OB or the value of the x coordinate of the point A.

Coordinates of A is ([math]cos\Theta,sin\Theta[/math]) given it is any point on the unit circle where theta is the angle formed between A, O, and the X-axis.[br][br]Cos theta = x[br]Sin theta = y[br]Sec theta = 1/x[br]Cosec theta = 1/y[br]Tan theta = x/y[br]Cot theta = y/x[br]where x and y are the coordinates of sin theta.[br]