 Hello and welcome to the session. In this session we will discuss derivation of the formula for the area of a triangle using trigonometric ratios. First of all we shall discuss area of a triangle. The area K of any triangle A, B, C is half the product of the length of its two sides and the sign of the included angle. Area K is given by 1 by 2 into B C sign of angle A is equal to 1 by 2 into A B sign of angle C which is equal to 1 by 2 into AC into sign of angle B. Now we shall see its derivation we know that area of any triangle is given by 1 by 2 into B into H where B is the base and H is the height of the triangle. This formula is used when we know the height H of the triangle. Let us consider a general triangle A, B, C. To remember the formula we always draw a triangle A, B, C generally such that the side opposite to angle A is of length B and side opposite to angle C is of length C. So here side opposite to angle A is B, C and its length is A, side opposite to angle B is AC and its length is B and side opposite to angle C is AB and its length is C. In this triangle we take BC as the base of the triangle for height. Let us draw a perpendicular line from vertex A to base BC meeting it at point D. Then perpendicular AD gives us the height H of the triangle. Now triangle ABD is a right angle triangle with hypotenuse AB of length C. We know that in a right angle triangle sign of angle theta is given by opposite side by hypotenuse. So in triangle ABD sign of B is equal to opposite side that is side AD which is of length H upon hypotenuse that is side AB which is of length C. So sign of angle B is equal to H upon C which implies that C into sign of angle B is equal to H. So we have H is equal to C sign B. Now we have got the value of height H and we know that area of triangle is given by half into base into height. So this is equal to 1 by 2 into base and here base is BC which is of length A into height and height is given by C sign B which is equal to 1 by 2 A into C into sign B. Now let us denote the area of triangle ABC by K. So this implies that K is equal to 1 by 2 into AC sign B. Now we see that angle B is the included angle between the two sides of the triangle of length A and C. Thus area K of any triangle say triangle ABC is one half of the product of length of any two sides and the sign of the included angle similarly we can take any pair of sides and the included angle to find area of the triangle. Now in this triangle ABC if we take AC as base and draw perpendicular BD from vertex B then from triangle ADB which forms a right angle triangle. Sign of angle A is given by opposite side that is H upon hypotenuse that is C which implies that H is equal to C into sin of angle A and here base AC is equal to B then area of triangle ABC which is given by K is equal to 1 by 2 into base that is B into height that is C into sin of angle A which implies that K is equal to 1 by 2 into BC sin A where we have taken length of sides AC and AB and the included angle between these two sides similarly we can also have area of triangle ABC that is K is equal to 1 by 2 into AB sin C where we have taken length of sides BC and AC and the included angle between these two sides we should note that we can use any of the three results to find the area of the triangle because all of them give area of same triangle ABC so they all are equal thus we have the area K of any triangle ABC is one half the product of the length of the two sides and the sign of the included angle that is area of the triangle given by K is equal to 1 by 2 into BC sin A which is equal to 1 by 2 into AB sin C which is equal to 1 by 2 into AC sin B let us consider an example find the area of the given triangle and here is the given triangle ABC here we are given length of two sides and the measure of the included angle between these two sides let us denote the length of side AC by B the length of side AB by C and the length of side BC by A so we have length of side AC that is B is equal to 15 centimeters length of side AB that is C which is equal to 10 centimeters and we are given angle A which is equal to 50 degrees now here we will use the formula that is area of triangle is given by 1 by 2 into BC sin of angle A now we put values of BC and angle A and we get 1 by 2 into B that is 15 into C that is 10 into sin of angle A that is sin of 50 degrees which is equal to 1 by 2 into 15 into 10 into sin of 50 degrees is equal to 0.76 we have calculated this value using calculator now simplifying it further this is approximately equal to 57 so area of triangle ABC is approximately 57 square centimeters does in this session we have learned the derivation of the formula for the area of a triangle using trigonometric ratios this completes our session hope you enjoyed this session