Class 11 Maths Chapter 12 Notes

• Definition of angle of elevation and Angle of Depression.
• Our comprehensive Class 11 Maths Chapter 12 Notes will ensure you're fully prepared for your exams.
• Solution of each question of Exercise 12.1 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.2 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.3 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.4 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.5 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.6 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.7 in 1st year Mathematics of the Punjab Textbook Boards.
• Solution of each question of Exercise 12.8 in 1st year Mathematics of the Punjab Textbook Boards.
• In the tables of Natural Sines, we get the number (nearest to 5100) 5090 which lies at the intersection of the row beginning with 30° and the column headed by 36’. The diference between 5100 and 5090 is 10 which occurs in the row of 30° under the mean diference column headed by 4’. So, we add 4’ to 30° 36’ and get Hence x = 30° 40.
• Solution of Right Triangles: In order to solve a right triangle, we have to ind: i) the measures of two acute angles and ii) the lengths of the three sides. We know that a trigonometric ratio of an acute angle of a right triangle involves 3 quantities “lengths of two sides and measure of an angle”. Thus if two out of these three quantities are known, we can ind the third quantity. Let us consider the following two cases in solving a right triangle:
• Height And Distances: One of the chief advantages of trigonometry lies in inding heights and distances of inaccessible objecst: In order to solve such problems, the following procedure is adopted: 1) Construct a clear labelled diagram, showing the known measurements. 2) Establish the relationships between the quantities in the diagram to form equations containing trigonometric ratios. 3) Use tables or calculator to ind the solution.
• Angles of Elevation and Depression: If OA is the horizontal ray through the eye of the observer at point O, and there are two objects B and C such that B is above and C is below the horizontal ray OA, then,  i) for looking at B above the horizontal ray, we have to raise our eye , and ∠AOB is called the Angle of Elevation and  ii) for looking at C below the horizontal ray we have to lower our eye , and ∠AOC is called the Angle of Depression. Example 1: A string of a lying kite is 200 meters long, and its angle of elevation is 60°. Find the height of the kite above the ground taking the string to be fully stretched. Solution: Let O be the position of the observer, B be the position of the kite and OA be the horizontal ray through O.
• Engineering and Heights and Distances: Engineers have to design the construction of roads and tunnels for which the knowledge of heights and distance is very useful to them. Moreover, they are also required to ind the heights and distances of the out of reach objects. Example 4: An O.P., sitting on a clif 1900 meters high, inds himself in the same vertical plane with an anti-air-craft gun and an ammunition depot of the enemy. He observes that the angles of depression of the gun and the depot are 60° and 30° respectively. He passes this information on to the headquarters. Calculate the distance between the gun and the depot. Solution: Let O be the position of the O.P., A be the point on the ground just below him and B and C be the positions of the gun and the depot respectively.