10. FACTS
Surveyors use two instruments, the transit and
the theodolite, to measure angles of elevation
and depression. On both instruments, the
surveyor sets the horizon line perpendicular
to the direction of gravity. Using gravity to find
the horizon line ensures accurate measures
even on sloping surfaces, industrial and
commercial buildings, when planning to set out
roads, driveways, retaining walls and site
grading.
11. You Need-to-Remember
Sin θ = Opposite / Hypotenuse
Cos θ = Adjacent / Hypotenuse
Tan θ = Opposite / Adjacent
To find an angle use inverse Trig Function
Trig Fnc-1 (some side / some other side) = angle
To Solve Any Trig Word Problem
Step 1: Draw a triangle to fit problem
Step 2: Label sides from angle’s view
Step 3: Identify trig function to use
Step 4: Set up equation
Step 5: Solve for variable Θ
Angle of Elevation
or of Depression
angle goes here
x
33°
25 y°
z
12. Example
Job Site A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet
from the base of the building. What angle does the ladder make with the ground?
x°
Step 1: Draw a triangle to fit problem
8
20
Step 2: Label sides from angle’s view
adj
hyp
Step 3: Identify trig function to use
S O / H
C A / H
T O / A
Step 4: Set up equation
8
cos x° = -----
20
Step 5: Solve for variable
cos-1 (8/20) = x
x= 66.42°
29. An angle of elevation is the angle formed by a
horizontal line and a line of sight to a point above the
line. In the diagram, 1 is the angle of elevation from
the tower T to the plane P.
An angle of depression is the angle formed by a horizontal
line and a line of sight to a point below the line. 2 is the
angle of depression from the plane to the tower.
Definition
30. Definition
Angle of Elevation
• The angle between a
horizontal line and the line
joining the observer’s eye to
some object above the
horizontal line is called the
angle of elevation.
31. Definition
Angle of Depression
• The angle between a
horizontal line and the line
joining the observer’s eye to
some object below the
horizontal line is called the
angle of depression.
•
32. Identifying Angles of Elevation
and Depression
1
2
depression
of
angle
the
is
1
elevation
of
angle
the
is
2
33. Example 1: Classifying Angles of Elevation and Depression
Classify each angle as an angle of
elevation or an angle of depression.
1 3
1 is formed by a horizontal line and a line of sight to a point
below the line. It is an angle of depression.
3 is formed by a horizontal line and a line of sight to a point
below the line. It is an angle of depression.
34. Example 2: Classifying Angles of Elevation and Depression
Classify each angle as an angle of
elevation or an angle of depression.
2 4
2 is formed by a horizontal line and a line of sight to a point above
the line. It is an angle of elevation.
4 is formed by a horizontal line and a line of sight to a point above
the line. It is an angle of elevation.
35. Example 2: Classifying Angles of Elevation and Depression
Classify each angle as an angle of
elevation or an angle of depression.
4
4 is formed by a horizontal line and a line of sight to a point above
the line. It is an angle of elevation.
36. LETS GO!!!!!
Use the diagram above to classify each
angle as an angle of elevation or angle of
depression.
3a. 5
3b. 6
6 is formed by a horizontal line and a line of sight to a point above
the line. It is an angle of elevation.
5 is formed by a horizontal line and a line of sight to a point
below the line. It is an angle of depression.
37. CHECK OUT THIS Example
A man looks out to sea from a cliff top at a height of 12 metres. He
sees a boat that is 150 metres from the cliffs. What is the angle of
depression
Solution
The situation can be
represented by the triangle
shown in the diagram, where θ
is the angle of depression.
Using
?
?
?
? to 1 decimal place
38. CHECK OUT THIS Example
A ladder is 3.5 metres long. It is placed against a vertical wall 1.75m
so that its foot is on horizontal ground is away from the wall.
(a)Draw a diagram which represents the information given.
(b)Calculate
(i) the angle of elevation from the ground to the top of the wall.
(ii) the angle of depression from the ground to the top of the wall.
5 m
3m
39. Check It Out! #1
What if…? Suppose the plane is at an altitude of
3500 ft and the angle of elevation from the tower to
the plane is 29°. What is the horizontal distance
between the plane and the Tower?
Simplify the expression.
40. Check It Out! #1 (cont)
What if…? Suppose the plane is at an altitude of 3500 ft and the angle
of elevation from the tower to the plane is 29°. What is the horizontal
distance between the plane and the Tower? (to the nearest ft)
Simplify the expression.
41. Check It Out! #1
SOLUTION
3500 ft
29°
You are given the side opposite A, and x is the
side adjacent to A. So write a tangent ratio.
Multiply both sides by x and divide by tan 29°.
x 6314 ft
Simplify the expression.
42. Check It Out! #2
What if…? Suppose the ranger sees another fire and the angle of
depression to the fire is 3°. What is the horizontal distance to this
fire? ( to the nearest foot)
By the Alternate Interior Angles Theorem, F = 3°.
Write a tangent ratio.
Multiply both sides by x and divide by tan 3°.
x 1717 ft Simplify the expression.
3°
43. Check It Out! #3
• You sight a rock climber on a cliff at a 32o angle
of elevation. The horizontal ground distance to
the cliff is 1000 ft. Find the line of sight
distance to the rock climber.
32
1000 ft
x
x
1000
32
Cos
32
Cos
1000
x
ft
1179
x
44. Check It Out! #4
• An airplane pilots sights a life raft at a 26o
angle of depression. The airplane’s altitude is
3 km. What is the airplane’s surface distance d
from the raft?
26
26
3 km
d
d
3
26
Tan
26
Tan
3
d
km
2
.
6
d
45. Check It Out! #2
Meteorology
One method that meteorologists could use to find the
height of a layer of clouds above the ground is to shine a
bright spotlight directly up onto the cloud layer and
measure the angle of elevation from a known distance
away.
46. Find the height of the cloud layer in the
diagram to the nearest 10 m.
Check It Out! #2 (cont)
Meteorology
47. Example 6: Finding Distance by Using Angle of Depression
An ice climber stands at the edge of a crevasse that is 115 ft
wide. The angle of depression from the edge where she
stands to the bottom of the opposite side is 52º. How deep is
the crevasse at this point?
48. Example 6 Continued
Draw a sketch to represent the given
information. Let C represent the ice
climber and let B represent the bottom
of the opposite side of the crevasse. Let
y be the depth of the crevasse.
49. Example 6 Continued
By the Alternate Interior Angles Theorem, mB = 52°.
Write a tangent ratio.
y = 115 tan 52° Multiply both sides by 115.
y 147 ft Simplify the expression.
50. Example 8: Shipping Application
An observer in a lighthouse is 69 ft above the water. He sights
two boats in the water directly in front of him. The angle of
depression to the nearest boat is 48º. The angle of depression
to the other boat is 22º. What is the distance between the two
boats? Round to the nearest foot.
51. Example 8 Application
Step 1 Draw a sketch. Let L
represent the observer in the
lighthouse and let A and B
represent the two boats. Let x
be the distance between the
two boats.
52. Example 8 Continued
Step 2 Find y.
By the Alternate Interior Angles Theorem, mCAL = 58°.
.
In ∆ALC,
So
53. Step 3 Find z.
By the Alternate Interior Angles Theorem, mCBL = 22°.
Example 8 Continued
In ∆BLC,
So
54. Step 4 Find x.
So the two boats are about 109 ft apart.
Example 8 Continued
x = z – y
x 170.8 – 62.1 109 ft
55. Check It Out! Example 9
A pilot flying at an altitude of 12,000 ft sights two airports
directly in front of him. The angle of depression to one airport
is 78°, and the angle of depression to the second airport is
19°. What is the distance between the two airports? Round to
the nearest foot.
56. Step 1 Draw a sketch. Let P
represent the pilot and let A and B
represent the two airports. Let x be
the distance between the two
airports.
Check It Out! Example 9 Continued
78°
19°
78° 19°
12,000 ft
57. Step 2 Find y.
By the Alternate Interior Angles Theorem, mCAP = 78°.
Check It Out! Example 9 Continued
In ∆APC,
So
58. Step 3 Find z.
By the Alternate Interior Angles Theorem, mCBP = 19°.
Check It Out! Example 9 Continued
In ∆BPC,
So
59. Step 4 Find x.
So the two airports are about 32,300 ft apart.
Check It Out! Example 9 Continued
x = z – y
x 34,851 – 2551 32,300 ft
61. Lesson Quiz: Part I
Classify each angle as an angle of elevation or angle of
depression.
1. 6
2. 9
angle of depression
angle of elevation
62. Lesson Quiz: Part II
3. A plane is flying at an altitude of 14,500 ft. The angle of
depression from the plane to a control tower is 15°. What is
the horizontal distance from the plane to the tower? Round to
the nearest foot.
4. A woman is standing 12 ft from a sculpture. The angle of
elevation from her eye to the top of the sculpture is 30°, and
the angle of depression to its base is 22°. How tall is the
sculpture to the nearest foot?
54,115 ft
12 ft
63. Problem 1
Julian is at the base of the building and he wishes to
know its height. He walks along to a point 90 ft from
the base of the building, and from that point he
measures the angle of elevation of the top of the
building to be 50°. What is the height of the
building? Round off answer to the nearest whole
number.
64. Solution
tan50
90
h
ft
tan50 (90 )
ft h
1.1918(90 )
ft h
107.26 ft h
Let h = height of the building
107 ft h
The height of the building is 107ft.
65. Problem 2
If a kite is 150 ft high and when 800
ft of string is out, what is the
measure of the angle does the kite
make with the ground?
67. Problem 3
• A 37 ft flag pole casts a 21 ft shadow.
What is the angle of elevation of the
sun? Round off your answer to the
nearest whole number and let theta be
the angle of elevation of the sun.
69. Problem 4
From the top of a 115 ft lighthouse,
the angle of depression of a boat on
the sea is 10°15’. Find the distance of
the boat from the base of the
lighthouse.
Let x be the distance of the boat from
the boat of the lighthouse
70. Problem 5
Aris stands 105 ft away from the
base of a tree. He measures the
angle of elevation to the top of
the tree to be 72°. How tall is the
tree?
Let h be the height of the tree.
