 Hello and welcome to the session. In this session we will discuss derivation of the law of science and its application to solve triangles. First of all we shall discuss law of science. Let triangle A, B, C be any triangle with A, B and C representing the measure of sides opposite to angles with measurement A, B and C respectively. Then sin of angle A upon A is equal to sin of angle B upon B is equal to sin of angle C upon C or we can write it as A upon sin of angle A is equal to B upon sin of angle B is equal to C upon sin of angle C. Now we shall discuss its proof which is very simple. In this triangle A, B, C let us draw a line perpendicular from vertex A meeting side B, C at point D then perpendicular A, D gives us the height H of the triangle. Now we see that triangle A, B, D is a right angle triangle with hypotenuse A, B of length C and perpendicular A, D of length H. So here sin of angle B would be equal to opposite side upon hypotenuse that is H upon C which implies that C into sin of angle B is equal to H or we can write it as H is equal to C into sin of angle B. We mark this equation as equation number one. Similarly in right angle triangle A, D, C sin of angle C would be equal to opposite side that is H upon hypotenuse that is B which implies that B into sin of angle C is equal to H or we can write it as H is equal to B into sin of angle C and we mark this equation as equation number two. Now from equation number one and equation number two we get C into sin of angle B is equal to B into sin of angle C which implies that sin of angle B upon B is equal to sin of angle C upon C and we mark this equation as equation number three. Now if we draw perpendicular B D prime from vertex B then from right angle triangle A D prime B we have sin of angle A is equal to H upon C which implies that H is equal to C into sin of angle A also in right angle triangle B D prime C we have sin of angle C is equal to H upon A which implies that H is equal to A into sin of angle C. Now we have H is equal to C into sin of angle A and H is equal to A into sin of angle C from these two equations we can write C into sin of angle A is equal to A into sin of angle C which implies that sin of angle A upon A is equal to sin of angle C upon C and we mark this equation as equation number four. So now from equation number three and equation number four we get sin of angle A upon A is equal to sin of angle B upon B is equal to sin of angle C upon C. If we take its reciprocal we can also write it as A upon sin of angle A is equal to B upon sin of angle B is equal to C upon sin of angle C. Hence a result is proved now we are going to discuss how to solve triangles using law of science to solve a triangle means to find the length of all its sides and the measure of all its angles Law of science can be used to write three different equations and these are sin of angle A upon A is equal to sin of angle B upon B or sin of angle B upon B is equal to sin of angle C upon C or sin of angle A upon A is equal to sin of angle C upon C. To solve any triangle we can apply the law of science if we know the three dimensions of a triangle and we need to find the other three dimensions. We should note that the three dimensions could not be any three dimensions. Now law of science is applicable when we are given AAS that is any two angles and an adjacent side. If we take these two angles then we can have any of these two sides ASA that is any two angles and their included side that is we can take any two angles and a side between these two angles. Next is SSA that is side side angle that is two sides and an angle. This is an ambiguous case. In this session we will only discuss these two cases that is AAS and ASA. We adopt the same method to solve a triangle when we are given AAS and ASA. Let us see the following example. In triangle ABC A is equal to 4.56 and then A is equal to 43 degrees and then C is equal to 57 degrees. We need to solve the triangle. First we have drawn this figure. This is triangle ABC with angle A as 43 degrees and then C as 57 degrees. Also side opposite to angle A is of length 4.56 and let us denote this length by A. Let side opposite to angle B is of length B and side opposite to angle C is of length C. So we know three of the six measures. We need to find angle B and length B and C. From this figure we see that we have AAS situation. So we begin by finding angle B. We know that sum of all angles of a triangle is 180 degrees. So here in triangle ABC we have measure of angle A plus measure of angle B plus measure of angle C is equal to 180 degrees. Which implies that measure of angle A that is 43 degrees plus measure of angle B plus measure of angle C that is 57 degrees is equal to 180 degrees. Which implies that 43 degrees plus 57 degrees is 100 degrees plus measure of angle B is equal to 180 degrees. Which implies that measure of angle B is equal to 180 degrees minus 100 degrees that is equal to 80 degrees. So measure of angle B is equal to 80 degrees. Now we need to find length of B and C. Using law of signs we have sin of angle A upon A is equal to sin of angle C upon C and sin of angle A upon A is equal to sin of angle B upon B. We first take this equation we get now sin of angle A that is sin of 43 degrees upon A that is 4.56 is equal to sin of angle C that is sin of 57 degrees upon C. Which implies that C is equal to 4.56 into sin of 57 degrees whole upon sin of 43 degrees. Now using calculator we find the values of sin of 57 degrees and sin of 43 degrees and we get C is equal to 4.56 into 0.83 whole upon 0.68. On solving further we get the value of C as 5.56 approximately. So C is approximately equal to 5.56. Now we shall solve this equation. Now we have sin of angle A upon A that is sin of 43 degrees upon 4.56 which is equal to sin of angle B that is sin of 80 degrees upon B. Which implies that B is equal to 4.56 into sin of 80 degrees whole upon sin of 43 degrees. Using calculator we find the values of sin of 80 degrees and sin of 43 degrees and we get B is equal to 4.56 into 0.98 whole upon 0.68. On solving further we get the value of B as 6.57 approximately. So B is approximately equal to 6.57. So we say that measure of angle B is equal to 80 degrees C is approximately equal to 5.56 and B is approximately equal to 6.57 which is the required answer. So in this session we have discussed the derivation of law of science and its application to solve triangles. This completes our session. Hope you enjoyed this session.