 Hello and welcome to the session. In this session we will discuss the application of laws of science and coscience. Laws of science and coscience are used in solving various real life problems like surveying problems, finding resultant force, finding distances, aviation problems etc. Let us consider the following example. Two boats A and B are sailing apart such that the angle between them is 72 degrees. Boat A sails for 6 miles and boat B sails for 11 miles. How far apart are the boats? Now let us draw its diagram. So we have boat A which has sailed for 6 miles and this is boat B which has sailed for 11 miles. And this angle formed is 72 degrees and let this be point C. And we have to find the distance between the two boats that is we have to find distance AB. Let us denote this distance by C. So we can see we have a triangle CAB with sides AC is equal to 6, BC is equal to 11 and angle C is equal to 72 degrees and we have to find side AB. So now we are given two sides and an included angle that is SAS is given. So we will use law of cosine equation that is C square is equal to A square plus B square minus of 2 into A into B into cos of angle C. Now we put the values of A which is equal to 11, B which is equal to 6, angle C which is equal to 72 degrees in this equation. And we get C square is equal to 11 square plus 6 square minus 2 into 11 into 6 into cos of 72 degrees. Which further implies that C square is equal to now 11 square is 121 plus 6 square that is 36 minus 2 into 11 into 6 that is equal to 132 into cos of 72 degrees. And using calculator we find the value of cos of 72 degrees as 0.309 this implies that C square is equal to 157 minus 132 into 0.309 that is equal to 40.78. Which further implies that C square is equal to 157 minus of 40.78 that is 116.22. Now taking positive square root we get the value of C as 10.78 approximately. So here C is approximately equal to 10.78 miles thus we can say that both are approximately 10.78 miles apart. Let us see one more example. A spectator watching a baseball game is sitting directly behind home plate in the last row of the upper deck of the stadium. The angle of depression to home plate is 31 degrees and the angle of depression to pitchers mount is 27 degrees. The distance between the home plate and the pitchers mount is 53.2 feet. How far is the spectator from the home plate? Let us draw its figure. Let point A be the home plate and point B be pitchers mount such that distance AB is equal to 53.2 feet. Let spectator be sitting at point C then this is the angle of depression to home plate and is equal to 31 degrees and this is angle of depression to pitchers mount and it is 27 degrees. We have to find distance AC and let us denote this distance by X. Now angle of depression is congruent to angle of elevation because they are alternate angles so we have angle B is equal to 27 degrees. Now CAB form a triangle. We can find angle C that is this angle which is given by 31 degrees minus 27 degrees that is equal to 4 degrees. So angle C is 4 degrees. Now we know angle B, angle C and side AB. So we are given two angles and adjacent side to one of the angles that is AAS. So we use law of signs to find X so we have sign of 4 degrees upon 53.2 is equal to sign of 27 degrees upon X. Now using calculator we find the value of sign of 4 degrees and sign of 27 degrees and here we have sign of 4 degrees that is 0.069 upon 53.2 is equal to sign of 27 degrees that is 0.453 upon X which implies that X is equal to 0.453 into 53.2 whole upon 0.069. And solving this further we get the value of X as 349.26 approximately. Thus we have got distance AC is equal to 349.26 feet. Thus we say that distance between the spectator and home plate is 349.26 feet approximately. Thus we can use these laws in real life. This completes our session. Hope you enjoyed this session.