lessons on Angle of Elevation and Depression

1. ACTIVITY
Look Up! Look Down!
Follow the steps below and answer the
questions that follow.
Use a tape measure/meter stick to
measure the distance between your eyes
and feet.
Move around the room and find an object
that is at the exact height as your eyes
and labels a picture.

2. Go outside the room and make an illustration
of:
i. Tall objects/structures
ii. Short objects/structures

3. Motive Question/Analysis
1. Describe the illustration or picture you
have created from the activity.
2. What mathematical concepts did you learn
from the activity? When you look up to tall
objects is there an angle formed? What
about when you look down?
3. Do you think you can directly measure the
height, the distance of the object you have
listed in the activity?

4. Enriching vocabulary
Line of sight – is an imaginary line that
connects the eye of an observer to the object
being observed.
Angle of Elevation – is the angle from the
horizontal to the line of sight of the observer to
the object above.
Angle of Depression – is the angle from
the horizontal to the line of sight of the
observer to the object below.

5. Guide Questions for the Discussion
a. How do you measure your height?
b. How do you measure tall objects
and structures?

6. Angle of Elevation
&
Angle of Depression

7. When a person looks at something above
his/her location, the angle between the
line of sight and the horizontal is called
the angle of elevation. In this case, the
line of sight is "elevated" above the
horizontal.
Eye
Object

8. When a person looks at something below
his or her location, the angle between the
line of sight and the horizontal is called
the angle of depression. In this case, the
line of sight is "depressed" below the
horizontal.
Eye
Object

9. Steps in solving the angle of elevation
and angle of elevation
Step 1: Draw a sketch of the situation.
Step 2: Mark in the given angle/leg of
elevation or depression.
Step 3: Use trigonometry to find the required
missing length/angle

10. A stands at the window of a
so that his are 12.6 m above the
level ground in the vicinity of the
An object is 58.5 m away from the
on a line directly beneath the
Find the angle of depression of the
person's line of sight to the object on the
ground.

11. Solutions
 = angle of depression
12.6tan
58.5
 
eye
12.6m
Ground level
object
58.5m

 1 12.6tan
58.5

 
  12.15º


12. Exercise
Calculate the angle of elevation of the
line of sight of a person whose eye is
1.7 m above the ground, and is looking
at the top of a tree which is 27.5 m away
on level ground and 18.6 m high.

13. Sketch

14. Solution
16.9tan
27.5
 
 1 16.9tan
27.5

 
  31.57º
Angle of elevation of this person line of
sight is 31.57º

15. Angle of depression
100m
h
Find the height of the man in the
hot air balloon from the ground.
= 50°
h =100 tan 50°
= 119.2m

16. FIND THE HEIGHT OF THE TOWER
80º
53m
Exercise
Height of tower
= 53 tan 80°
= 300.6m

17. Exercise
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?

18. Solution
Let x m be the distance of the object
from the base of the cliff.

19. APPLICATION
A man who is 2 m tall stands on horizontal
ground 30 m from a tree. The angle of elevation of
the top of the tree from his eyes is 28˚. Estimate
the height of the tree.

20. GENERALIZATION
 The study of trigonometric ratios
originated from geometric problems
involving triangles. Solving a triangle means
finding the lengths of the sides and
measures of the angles of the triangle.
Trigonometric ratios may be used to solve
problems involving angles of elevation and
depressions.

21. A trigonometric ratio often helps us set up
an equation, which can then be solved for
the missing measurement. If two legs of
the triangle are part of the problem, then it
is a tangent ratio. If the hypotenuse is part
of the problem, then it is either a sine or
cosine ratio

22. EVALUATION
1. Find the value of the hypotenuse x.
2. Find the measure
of angle x.

23. ASSIGNMENT
a. Find the height of a hot air balloon
b. Find the measure of
angle x above the ground