Download Angles of Elevation

Essential Math 30S
Trigonometry — Lesson 2
Angles of Elevation
The angle of elevation is the angle between a horizontal line from the observer
and the line of sight to an object that is above the horizontal line.
In the diagram below, AB is the horizontal line. The angle from the observer at A
to the object at C is called the angle of elevation.
(Object)
(Angle of elevation)
A
(Observer)
Horizontal line
"When you solve problems like this, the best thing to do is to draw it."
-
Trigonometry — Angles of Elevation
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Elaine Janz
Essential Math 30S
Trigonometry — Lesson 2
Example
A campsite is 9.41 km from a point directly below the mountain top. If the angle
of elevation is 12° from the camp to the top of the mountain, how high is the
mountain?
Solution
Sketch it first.
mountain top
ogle of elevation
941 km
You have been given the length of the adjacent side and you need to find the
length of the opposite side, so you will use the tan ratio.
Tan 12 =— 0.213= —
9.41
9.41
0.213* 9.41 = x
The mountain is 2.0 kms high.
Trigonometry—Angles of Elevation
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x = 2.0 kms
Essential Math 30S
Trigonometry -- Lesson 2
Assignment
A building casts a shadow of 24 m. If the angle of elevation from the end of
the shadow to the top of the building is 21°, find the height of the building.
2.
A building is 30 meters tall. Jane is standing at a point where the angle of
elevation is 37°. How far away is she standing from the building?
Trigonometry — Angles of Elevation
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Essential Math 305
Trigonometry — Lesson 2
3.
From a point 45 meters from the base of a church, the angle of elevation of
the top of the building is 27°.
a)
How tall is the building?
b)
There is a cross at the top of the church. If the angle of elevation
from that same point is 35°, what is the height of the cross?
Trigonometry— Angles of Elevation
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Essential Math 30S
Trigonometry — Lesson 2
.
A ramp is 75 feet long. The angle of elevation of the ramp is 12°. How high
does the ramp go?
2.°
Trent, who is 1.60 m tall, is standing in front of a tree. At point x, the angle
of elevation to Trent is 15° and the angle of elevation to the top of the tree
is 50°.
-Haw far is Point x from the tree?
b)
How high is the tree?
Trigonometry — Angles of Elevation
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Essential Math 30S
Trigonometry — Lesson 2
6.
Denton is standing half way between 2 buildings that are 50 meters apart.
The angle of elevation to the one building is 300 and the other angle of
elevation is 47°. How tall are the buildings?
2.5
Trigonometry — Angles of Elevation
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Essential Math 305
Trigonometry — Lesson 3
Angles of Depression
The angle of depression is the angle between a horizontal line from the observer
and the line of sight to an object that is below the horizontal line.
In the diagram below, PQ is the horizontal line. The angle from the observer at P
to the object at R is the angle of depression.
Horizontal line
9 (Angle of depression)
R (Object)
Trigonometry—Angles of Depression
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Essential Math 305
Trigonometry — Lesson 3
Example
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 34°. How far is the object from the base of the
cliff?
Solution
Let x meters be the distance of the object from the base of the cliff.
Top of cliff Ci
Base of Cliff A
P(Object)
X
in
Did you notice that...
When you have been given the angle of depression, it will also be the angle of
elevation! So you know that angle P is 34°. You have been given the length of the
opposite side and you want to find the adjacent side. You need to use the
tangent ratio.
40
Tan 34 = —
x
40
0.675 = —
x
0.675x
0.675x = 40
40
0.675 — 0.675
x = 59.2 or 59 m
The object is 59 m away from the base of the cliff.
Trigonometry — Angles of Depression
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Essential Math 30S
Trigonometry — Lesson 3
Example
A helicopter is observing a group of campers. If the angle of depression is 29°
and the helicopter is a horizontal distance of 250 meters from them, how far
above the ground is it?
Solution
Once again, you need to draw a sketch.
Angie of depression 29°
Helicopt
250
campers
.
If the angle of depression is 29°, then the alternate angle is also 29°. Label the
sides of the triangle. The height becomes the opposite side of the angle. The side
of the triangle that is 250 meters in length is the adjacent. Therefore we will use
the tan ratio.
Tan 29° = °
27
0 0.554 250 = 138.57 meters above the ground.
Trigonometry — Angles of Depression
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Essential Math 305
Trigonometry — Lesson 3
Assignment
1.
Calculate the angle of depression for each of these triangles.
tk,4
a)
12
b)
2.
Tracy is at the edge of a cliff and throws a rock to a point 40 meters from
the base of the cliff. If the cliff is 60 meters high and Tracy is 1.5 meters
tall, what is the angle of depression?
Trigonometry —Angles of Depression
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Essential Math 305
Trigonometry — Lesson 3
A man stands at the top of a 50 meter lighthouse and sees a boat. If the
angle of depression to sight the boat is 42°, find the distance between the
base of the lighthouse and the boat.
A ramp has a height of 4 feet and an angle of depression of 200 , What is
the length of the ramp?
hi?
Trigonometry— Angles of Depression
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Essential Math 30S
Trigonometry — Lesson 3
5.
A man stands on top of a cliff and sees a mountain goat on a hill at an angle
of depression of 400. If the hill and the cliff are 150 meters apart, how
much higher is the cliff than the hill?
Trigonometry —Angles of Depression
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