 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that A man sees a 440 feet tall pole from the roof of a building. The angle of elevation while viewing the top of the pole is 63 degrees and the angle of depression while viewing the base of the pole is 42 degrees. Find the height of the building. Before starting the solution of this question we should know our result. Law of science states that let triangle A,B,C be any triangle with A,B and C representing the measure of sites opposite to angles with measurements 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. With this key idea we shall proceed to the solution. In this question we are given that a man standing on the roof of a building sees a tall pole of height 440 feet. The angle of elevation while viewing the top of the pole is 63 degrees and the angle of depression while viewing the base of the pole is 42 degrees. We have to find the height of the building. Let us make its figure. This is the pole that is 440 feet tall. This is the building. Let it be x feet tall. The man is standing on the roof of the building. The angle of elevation to top of the pole is 63 degrees and angle of depression to base of the pole is 42 degrees. Let this be point A, this be point B, this is point C and this be point D. We have to find length AD which is equal to x. Now angle ACD will be equal to the angle of depression that is equal to 42 degrees as they form pair of alternate angles. Now we see that triangle ADC is a right angle triangle. Now we know that sin of angle theta is equal to opposite side upon hypotenuse. So here sin of 42 degrees will be equal to opposite side that is x upon hypotenuse AC which implies that x is equal to AC into sin of 42 degrees. We mark this equation as equation number 1. So to find the height of the building we need to find side AC. Now to find the length of side AC we need to find angle B. From the figure we can see that angle BAC is equal to 63 degrees plus 42 degrees which is equal to 105 degrees. Now since angle BCD is equal to 90 degrees angle BCA is equal to angle BCD minus angle ACD that is equal to angle BCD which is equal to 90 degrees minus angle ACD that is equal to 42 degrees. So we have 48 degrees so angle BCA is equal to 48 degrees that is this angle is equal to 48 degrees. So now in triangle ABC we have angle A which is equal to 105 degrees, angle C is equal to 48 degrees and we need to find angle B. Now using angle sum property that is sum of all angles of a triangle is 180 degrees we have angle A plus angle B plus angle C is equal to 180 degrees which implies that 105 degrees plus angle B plus 48 degrees is equal to 180 degrees which implies that 105 degrees plus 48 degrees that is 153 degrees plus angle B is equal to 180 degrees which further implies that angle B is equal to 180 degrees minus 153 degrees which implies that angle B is equal to 27 degrees. So now we have got the measure of angle B as 27 degrees. Now in triangle ABC we have side BC let us denote its length by A and this is equal to 440 feet that is A is equal to 440, angle B is equal to 27 degrees, angle BAC that is this complete angle is equal to 105 degrees. So we say that angle angle side situation is given that is AAS is given. So now using law of signs we can find the length of side AC let us denote this length by B. Now using law of signs we have sign of angle BAC that is sign of 105 degrees upon A that is 440 is equal to sign of angle B that is sign of 27 degrees upon B. Now this implies that B is equal to 440 into sign of 27 degrees whole upon sign of 105 degrees. Now using calculator we find the values of sign of 27 degrees and sign of 105 degrees and we get B is equal to 440 into 0.45 whole upon 0.96. On solving further we get the value of B as 206.2 which is approximately equal to 206. So now we got the value of length of side AC as 206. Now we put this value that is we put AC is equal to 206 in equation number 1 that is X is equal to AC into sign of 42 degrees and we get X is equal to AC that is 206 into sign of 42 degrees. And using calculator we get the value of sign of 42 degrees as 0.66. So this implies that X is approximately equal to 135.96 that is approximately equal to 136. So we say that height of building is approximately 136 feet. This is the required answer. This completes our session. Hope you enjoyed this session.