Applications of Trig

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Imogene Chase
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1 Applications of Trig We have to define a few terms: angle of Elevation, E, and angle of Depression, D. Both are measured from the horizontal line of sight. horizontal & D : Angle of Depression E : Angle of elevation horizontal 1) Frank is standing 10m from a tree. He estimates that the angle of elevation to the top of the tree is 60. Calculate the height of the tree? Please round off to the nearest whole number. Solution Sketch a diagram. tan 60 h 10 h 10tan60 h h 10 m The tree is 17m high.

2 2) Two buildings are directly across the street from each other. The distance between the front doors is 25m. Ana looks through a window in the taller building. She measures the angle of depression to the base of the shorter building. She records a value of 23. She also measures the angle of elevation to the top of the shorter building. She records a value of 31. Calculate the height of the shorter building to the nearest metre. Solution Once again sketch a diagram. It is just to assist with the solution, so don t worry about perfection b a 25m a b tan 23 tan tan 23 a 25 tan31 b 10.6 a 15.0 b The shorter building is 26m high. a b 25.6

3 Now, I want to consider what is known as the ambiguous case: Consider a base line as shown... The length is not known. Now another side of a known length is added to the left point at some fixed angle as shown... Now a third side of a known length is added to the top and we have two possibilities...which we call The Ambiguous Case...meaning there is more than one possibility. Here is one possibility... And here is another...

4 We have 2 different looking triangles that satisfy the conditions...the Ambiguous Case. If we put both triangles together we end up with this... The Ambiguous Case is a situation that I want you to consider. This is just an example to help you understand that Math isn t just right or wrong...as you head into higher levels of Mathematics there are many situations that will have more than one solution. At times it is up to you to decide which solution will satisfy the given conditions...perhaps all solutions will satisfy the conditions, perhaps only some will or perhaps none will. The applications that we have seen are not necessarily difficult. The key is a neat diagram to represent the situation. Now we will consider an application of the Ambiguous Case on the next page... By the way, I left it for the next page so that the problem, the diagram and the solution can be left together for easy reference.

5 3) A rotating beacon (also known as a search light) can illuminate up to 100m. It is floating offshore at a public beach. A point on the beach is 225m from the beacon. The line of sight from that point to the beacon forms an angle of 21 with the shoreline. Determine the length of the beach that can be effectively illuminated by the beacon. Please express your answer to the nearest metre. Point on the beach P Solution The diagram is the key to the solution B R C d Rotating beacon 100 (the distance that must be calculated) A shoreline Study the diagram carefully...it is important to understand its construction. Then, calculate the unknown values for angle A, angle B, angle C and side d, in that order. sin A sin 21 First, consider the largest triangle: apply the Sine Law sin 21 sin A A 54 Now the smaller triangle, on the right, has 2 equal sides (isosceles) 2 equal angles A & B. A B 54. Recall that the sum of the angles in a triangle is C 180 C 72 d 100 Now, in that smaller triangle on the right, apply the Sine Law sin 72 sin m of the beach is effectively illuminated. 100 sin 72 d 118 sin 54

6 Here are some word problems for applications in Trigonometry: 1) A ladder 5m long leans against a vertical wall. It forms an angle of 65 with the ground. a) How high up the wall does the ladder reach? b) How far is the foot of the ladder from the base of the wall? 2) Airport A is 150km east of airport B. A plane is240km from airport B and 23 north of due west from airport B. How far is the aircraft from airport A, to the nearest km? 3) The Toronto Stock Exchange is located in the Exchange Tower. From the top of the building, the angle of depression of a point on the ground 100m from the foot of the building is 56. Determine the height of the building, to the nearest metre. 4) An offshore rotating beacon can illuminate effectively up to 165m. A point on the beach is 350m from the beacon. The sight-line to the beacon, from that point, forms an angle of 20. Determine the length of the beach that can be effectively illuminated by the beacon, to the nearest metre. 5) Two ships leave Port Hope on Lake Ontario, at the same time. One travels at 12km/h on a course of 235. The other travels at 15km/h on a course of 105. How far apart are the ships after four hours? Please answer to the nearest km. Use the diagram below to help you solve the problem. N W 270 E S 180

7 The trick to success is to draw a neat sketch to help you visualize the problem. Also, keep it as simple as possible...break it down to basics. In other words, take something that seems complicated and turn it into a simpler concept. Answers 1a) 4.5 1b) 2.1 2) 383 3) 148 4) 225 5) 98

Date: Worksheet 4-8: Problem Solving with Trigonometry

Date: Worksheet 4-8: Problem Solving with Trigonometry Worksheet 4-8: Problem Solving with Trigonometry Step 1: Read the question carefully. Pay attention to special terminology. Step 2: Draw a triangle to illustrate the situation. Decide on whether the triangle