SAMPLE LESSON: MATHEMATICS
Class: Form 5
Title of Module: Plane Geometry Title of Chapter: Trigonometry
Title of Lesson: Angle of Elevation, Angle of Depression
Duration of Lesson: 50mins
MATHEMATICS LESSON
CLASS: Form 5; Duration: 50 minutesTOPIC: Plane Geometry Lesson: Trigonometry
Lesson Objectives: At the end of the lesson, you should be able to: - Translate a given situation into a Mathematical figure;
- Find angle of Elevation; - Find angle of Depression; -. Find length of a distance
Prerequisite knowledge: You can do the following:
- Use Pythagoras theorem to solve a right triangle;
- Use appropriate trigonometric ratio to find lengths of sides and measures of angles in a triangle
If not go back and revise these notions.
Motivation: Practical applications of trigonometry by Engineers, Surveyors, Pilot and Navigators
frequently involve measure of angles of elevation and depression based on the given situation.
Angles of elevation and depression are often used in trigonometry word problems.
Every time you look up at something in the sky, you are creating something called the Angle
of Elevation with your eyes and when you look down at something on the ground, you are
creating an angle called Angle of Depression. Didactic Materials:
-Ruler, Pencil, calculator, References:
1. GEOMETRY, Eugene D. Nicholas, Mervine L. Edwards, E Henry Garland, Sylvia, A Hoffman, Albert Mamary, William F Palmer (1991), Holt, Rinehart and Winston, Inc.
2. https://www.mathsteacher.com.au/year10/ch15_trigonometry/12_elevation_depression/23elevdep.htm
Introduction
As earlier mentioned in the motivation, surveyors, pilot, navigators etc. usually apply trigonometric ratios to problems involving angles of elevation and depression as you will see with some examples in this lesson.
Verification of Pre-Requisite Knowledge
1. Find the value of x, in cm as indicated in the figure below.
2. Find the measure of the angle indicated as xo in the figure below.
3. The length of the altitude to the base of an isosceles triangle is 16m. The measure of a base angle is 55o. Find the length of the base of the triangle to the nearest meter.
4. Find the length indicated as x on the diagram below.
Answers: 1) x = 24.0m; 2) xo = 54o; 3) The length of the base = 22.4m; 4) x = 21.4
NB: If you have any of them wrong, go back and revise the notions before you continue the lesson.
Angle of Elevation Angle of Depression
Examples:
The height of the tree is BC. A man standing at a distance AC from the see will have to lift up his eyes to see the top of the tree. The angle through which he takes up his eyes, is the angle of
I- Angle of Elevation
If a person stands and looks up at an object, the angle of elevation is the angle between the horizontal line of sight and line through which the person sees the object.
This can be represented by the figure:
II- Angle of Depression
If a person stands and looks down at an object, the angle of depression is the angle between the horizontal line of sight and the line through which the person sees the object.
The diagram below summaries
In the diagram above, angle labelled 1 indicates the angles of elevation. It is the angle by which the ground observer’s line of vision must be raised or elevated with respect to the horizontal, to sight an object at B.
While the angle labelled 2 is the angle of depression. It is the angle by which an observer at B’s line of vision must lowered or depressed, with respect to the horizontal to sight an object at A.
Examples:
1. From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34º. How far is the object from the base of the cliff?
Let x m be the distance of the object from the base of the cliff.
Angle of depression = 34o
But 𝐴𝑃𝑂 ̂ = 𝐵𝑂𝑃̂ because they are alternate angles ∴ 𝐴𝑃𝑂 ̂ = 34𝑜 From triangle APO, we have: 𝑡𝑎𝑛34𝑜 = 40
𝑥 ∴ 𝑥𝑡𝑎𝑛340 = 40
𝑥 × x ⟹ 0.6745𝑥 = 40 ∴ 𝑥 = 59.30
2. You are on a trip through a desert. At a distance d, you can see mountains, and quick measurement tells you that the angle between the mountain top and the ground is 13.4o. You know that the highest point (the centre) of the mountain is 2500m high. How far away are you from the centre of the mountain?
tan 13.4o = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = 2500 𝑑 ⟹ 𝑑 = 2500 𝑡𝑎𝑛13.4𝑜 ≅ 10494𝑚
III- Application Exercises to find distances and angles
1. A plane is flying over level ground at an altitude of 900m. When the pilot sights a landing field, the measure of the angle of depression is 270.
a) Represent the above information on a diagram.
b) Find the distance, to the nearest meter, from the point on the ground directly under the pilot to the landing field (Ground distance).
Solutions:
a. Let F be the field for landing.
let P be the point the pilot sights the landing field and G the point on the ground directly under the pilot to the landing field.
b. The figure in a) is a right triangle. One of the trigonometric ratios can be used to find the ground distance.
On the diagram the length of the ground distance FG is the side adjacent to angle 27o while the side PG with distance of 900m is the opposite side.
tan 27 = 𝑃𝐺
𝐺𝐹 = ⟹ 0.5095 = 900
𝐺𝐹 (Your calculator should give you Tan 27
0 = 0.5095 to
4decimal places)
∴ 𝐺𝐹 = 900 × 0.5095 = 1766.4𝑚 𝑻𝒉𝒆 𝑮𝒓𝒐𝒖𝒏𝒅 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = 𝟏𝟕𝟔𝟔𝒎
2. A tree 50m high casts a 35m shadow.
Solution:
The situation can be represented by the figure by the side.
From the figure, and using trigonometric ratios, 𝑡𝑎𝑛𝐴 = 50 35 ⟹ 𝑡𝑎𝑛𝐴 = 1.4285 ⟹< 𝐴 = 𝑡𝑎𝑛−11.4285 ∴ 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑛 ≅ 55𝑜 Points to remember
The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).
The angle of elevation of the object from the observer is 𝛼
If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.
The angle of depression of the object from the observer is 𝛽.
Home work
Do the following, show all figures, if not given and all necessary steps in your working.
1. A 20m ladder is leaning against a wall. The foot of the ladder forms an angle of 65o with the ground.
Determine, to the nearest meter, how far, the top of the ladder is from the ground.
2. A navigator at a point A observes the measure of the angle of elevation to the top of the cliff to be 15o. This measure changes to 26o at B as in the figure below. If the height of the cliff is known to be 750m. Find how far the boat moved to get from point A to point B.
3. A surveyor wants to measure the distances from points A and B to point C on the opposite sides of a stream. Point C can be sighted from both A and B. she measured AB and angles at A and B as indicated in the diagram. Find AC and BC to the nearest metre.
4. You are standing 10 meters away from a tree. The angle of elevation from your eyes to the top of the tree is 65o. Find how far away you are from the tree given that the distance from your feet to your eyes is 1.6m tall.
