Let ABC be a right angled triangle with angle ABC equal to 90 degrees. Let angle BCA be theta and it follows that angle BAC is equal to 90 – theta as the sum of the angles of a triangle is equal to 180 degrees.

Sin(theta) is defined as opposite side/hypotenuse and cos(theta) is defined as adjacent side/hypotenuse. Here the Adjacent side is BC, Opposite side is AB and the hypotenuse is AC.

Sin(Theta) = AB/AC; Cos(Theta) = BC/AC.. (1)

The pythogoreas theorem states that the square of the lengths of the opposite side and the adjacent side is equal to the square of the length of the hypotenuse.

So AB * AB + BC * BC = AC *AC – (2)

AB = AC * Sin(Theta) BC = AC * Cos(Theta)

So the RHS of the expression (2) can be rewritten as

AC * AC * Sin(Theta) * Sin(Theta) + AC * AC * Cos(Theta) * Cos(Theta). (3)

It follows from (2) and (3) that

(Sin(Theta) * Sin(Theta)) + (Cos(Theta) * Cos(Theta)) = 1.

This is the most fundamental identify in Trigonometry. All other identities follow from this.

Let us derive some more properties of trigonometric ratios.

Let us take angle BAC for consideration. Let us write down the expression for Sin(BAC) is equal to AB/AC =Cos(ABC). If Angle ABC = theta, then BAC = 90 – Theta. So Sin(90-theta) is equal to Cos(Theta) or in other words Sin(30) = Cos(60), Sin(60) is equal to Cos(30).

Let us now derive some more trigonometric identities such as Sin (A)/a, Sin (B)/b, Sin(C)/c etc.

Let Angle A be CAB, Angle B be ABC, Angle C be BCA

Sin (BCA) = AB/AC; Sin(CBA) = BC/AC.

Hence AB/Sin(BCA) = AC; BC/Sin(CBA) = AC;

So AB/Sin(BCA) = BC/Sin(CBA) implying that the ratio of Sin of an angle in a right angled triangle to the opposite side is equal. Now It has to be proved that this ratio also holds true for any other triangle, not only a right angled triangle. This can be done by drawing a perpendicular in case of non right triangles and deriving expressions for a/Sin(A), b/Sin(B) using two right triangles.

The values of trigonometric ratios such as sin, cos and tan for different values theta such as 30,45, 60 and 90 degrees Sin(30) can be found using the property of an isosceles /equilateral triangle.

An isosceles right triangle has two angles equal to 45 degree. If each side is equal to 1 the hypotenuse is equal to sqrt(2). So it follows that Sin(45) is equal to 1/sqrt(2).