Secondary 3 Mathematics Chapter 10 Applications of Trigonometry Practice 1 Learning Objectives: To provide an aim for

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1 Secondary 3 Mathematics Chapter pplications of Trigonometry Practice 1 Learning Objectives: To provide an aim for students to achieve at the end of each lesson. Understand and solve problems involving angle of elevation and angle of depression. Use trigonometric ratios, sine rule and cosine rule to find bearings between 2 points. Practical pplications of Trigonometry Trigonometry can be applied in real life situations to find height of mountains, the distance the shore is away from a point in the sea, the height of trees, distance between planets etc. Situation 1 : Find the height of the tree. Real-life examples are given clinometer is a simple instrument which is used to measure the angle to relate of a mathematics slope. You can to everyday life. create your own simple clinometer with a protractor, a straw, a pin, string and an eraser Look through the straw to focus at a point on the top of the tree, get another friend to read off the angle from the protractor. y is the vertical distance between the eye and the ground. θ distance away from tree tanθ= x distance away from tree x = tanθ Height of tree = x + y distanceawayfromtre x y -1-

2 Situation 2 : Find the width of the river. C Width of River θ x The width of a river can be found by not crossing the river itself. Start from point, look at a fix point across the river, point C. Then move yourself x m away from point to point. Find C with the help of a compass. width of river tanθ= x width of river = x tanθ ngles of elevation and depression n angle of elevation is the angle formed between the horizontal (eye level) and the line of sight of a point or an object above the horizontal. n angle of depression is the angle formed between the horizontal (eye level) and the lines of sight of a point or an object below the horizontal. Visuals help our students to quickly grasp the key concepts. ngle of depression ngle of elevation Note The angle of elevation from the ground to the top is equal to the angle of depression from the top to the ground. -2-

3 Example 1 The diagram shows a cliff PQ of height 52 m of which the Q angles of depression of Ship and Ship due east of it from Q are 36 and 24 respectively. Find the distance between the ships. P 52 m Solution: P =, P = tan36 tan Distance between 2 ships = tan24 tan36 = ( ) 45.2 m 3 s.f. Example 2 In the diagram, WX and YZ are 2 towers standing adjacent to each other. The angle of elevation of Y from W is and the angle of depression of Z from W is 22. Given that the height of WX is 32 m high, find a) the horizontal distance between the 2 towers, b) the height of YZ, c) the angle of depression of X from Y. W m X Y Z s much as we would love to show you everything, we cannot be showing you the best. Do drop by any JustEdu centre to view the full set! -3-

4 Exercise 1 1) In the diagram, Lisa is standing at point X. The angle of elevation of the top of a m tall tree from X is o. s she moves a distance of y m away from X, her new distance from the top of the tree is 18.3 m. The vertical distance between Lisa s eye level and the ground is approximately 1.6 m. Find a) the length of y, b) the new angle of elevation of the top of the tree from her new position. y m 18.3 m X m 2) t ground level, Willy stands m away from his HD block. The vertical distance between Willy s eye level and the ground is 1.75 m. The angles of elevation of the unit he is staying and the top of the block from his current position are 53 o and 65 o respectively. Find a) the height of his unit, b) the height of the HD block. 53 m 65 3) In the diagram, the distance between the top of building to the top of building is 86 m. The angle of elevation of the top of building from building is 35 ο If the height of building is 56 m, calculate a) the horizontal distance between the 2 buildings, b) the difference in height between the 2 buildings, c) the distance from the bottom of building to the top of building, d) the angle of depression of the bottom of building from the top of building. 86 m m 4) grey eagle is perched on top of a.2 m tall tree. brown eagle is on the ground, some distance away from the bottom of the tree. In between them stands a 4.8 m tree. t one instant, both eagles saw a prey on top of this tree. The angle of depression of the prey from the grey eagle is 43.8 and the angle of elevation of the prey from the brown eagle is If both eagles fly towards the prey in a straight path at the same speed, which eagle will get its meal? -4-

5 5) The angle of elevation of the foot of a lighthouse from a ship is 21. The lighthouse is on Do drop by our centre to view the full set of materials. earings earings are angles used to determine the position of points with reference to one another. earings are N ( ) measured relative to the North direction; read in the clockwise direction from the North; expressed as a 3-digit number; In the diagram, the bearing of from O is and the bearing of from O is 1. (2 ) W O 1 E (9 ) S (1 ) Example 3 Using the diagram as shown, state the bearing of a) from O, b) from O, c) C from O. Solution: a) earing of from O= C N b) earing of from O= 1 45 = 135 W 25 O E c) earing of C from O= = S

6 Example 4 In the figure, O, and are 3 checkpoints in a navigation exercise. Given that the bearing of from O is, is due northwest of O, = m and O = m, calculate the bearing of m a) O from, N b) O from, c) from, 45 m d) from. O Do drop by our centre to view the full set of materials. -6-

7 Example 5 Two ships P and Q leave a port R at the same time for their anchoring point. oth ships took 2 hours to reach their respective points with Ship P sailing at 12 km/h on a bearing of 2 and Ship Q sailing at km/h on a bearing of. a) Calculate their distance apart and the bearing of P from Q. b) nother Ship S has an anchoring point such that it forms a straight line with Ship P and Q. i) Calculate the bearing at which ship S should travel such that the distance from R to S is the minimum. ii) Hence, calculate this distance RS. Do drop by our centre to view the full set of materials. -7-