Transcript Geometry Section 10.3 Notes

10
CHAPTER
10.3 Solving Right
Triangles
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Slide 9-1
Solving Right Triangles
The process of determining the three angles and the
lengths of the three sides of a triangle is called
solving the triangle.
A right triangle can be solved if we know either
• the lengths of two sides or
• the length of one side and the measure of one acute
angle
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Slide 9-2
Solving a Right Triangle Given
Example
One Side and One Angle
Solve the right triangle. If needed, round any
answers to one decimal place.
Solution
m∠B can be found since ∠A and ∠B
are complements.
m∠B = 90° − m∠A = 90° − 32° = 58°
Use either angles A or B in the trigonometric ratios
to solve the rest of the triangle.
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Slide 9-3
Solving a Right Triangle Given
Example
One Side and One Angle
adj.
cos A 
hyp.
15
cos32 
c
c  cos32  15
15
c
cos32
15
c
0.8480
opp.
tan A 
adj.
a
tan 32 
15
15tan 32  a
15(0.6249)  a
9.4  a
mB is 58 degrees and a is
c  17.7
approximately 9.4 units and
c is approximately 17.7
units.
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Slide 9-4
Solving a Right Triangle Given
Example
Two Sides
Solve the right triangle. If needed,
round any answers to one decimal
place.
Solution
To find f, we can use the Pythagorean Theorem.
2
2
2
a b c
106  f
2
2
2
9 5  f
10.3  f
2
81  25  f
106  f
2
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Slide 9-5
Solving a Right Triangle Given
Example
Two Sides
Next, find m∠D or m∠E. We’ll
choose m∠D.
opp.
tan D 
adj.
9
tan D 
5
9
D  tan  
5
mD  60.9
1
∠E and ∠D are
complements:
m∠E = 90° − m∠D
≈ 90° − 60.9° = 29.1°
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Slide 9-6
Angles of Elevation and Depression
Many applications of right triangle trigonometry
involve the angle made with an imaginary horizontal
line. As shown in the figure below, an angle formed
by a horizontal line and the line of sight to an object
that is above the horizontal line is called the angle
of elevation. The angle formed by a horizontal line
and the line of sight to an object that is below the
horizontal line is called the angle of depression.
Transits and sextants are instruments used to
measure such angles.
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Slide 9-7
Angles of Elevation and Depression
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Slide 9-8
Identifying Angles of Elevation
Example
and Depression
What is a description of the angle as it relates to the
situation shown?
a. 1
b. 4
Solution
a. ∠1 is the angle of depression
from the bird to the person in the hot-air balloon.
b. ∠4 is the angle of elevation from the base of the
mountain to the person in the hot-air balloon.
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Slide 9-9
Problem Solving Using an Angle of
Example
Elevation
Sighting the top of a building, a surveyor measured
the angle of elevation to be 22°. The transit is 5 feet
above the ground and 300 feet from the building.
Find the building’s height to the nearest whole foot.
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Slide 9-10
Problem Solving Using an Angle of
Example
Elevation
Let a be the height of the part of the building that
lies above the transit. The height of the building is
the transit’s height, 5 feet, plus a.
The trigonometric ratio that will make it possible to
find a is tangent. In terms of the 22° angle, we are
looking for the side opposite the angle. Also, the
side adjacent to the 22° angle is 300 feet, and
tangent’s ratio is opp./hyp.
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Slide 9-11
Problem Solving Using an Angle of
Example
Elevation
opp.
tan 22 
a  300 tan 22
adj.
a  121
a
tan 22 
300
The height of the part of the building above the
transit is approximately 121 feet.
The height of the building is
h ≈ 5 + 121 = 126
The building’s height is
approximately 126 feet.
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Slide 9-12
Determining the Angle of
Example
Elevation
A building that is 21 meters tall casts a shadow 25
meters long. Find the angle of elevation of the sun to
the nearest degree.
Solution
opp.
tan A 
adj.
21
tan A 
25
1 21
A  tan
25
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Slide 9-13
Determining the Angle of
Example
Elevation
We use a calculator in the degree mode to find m∠A.
Tan-1
21
÷
25
Enter
The display should show
approximately 40. The angle
of elevation of the sun,
m∠A, is approximately 40°.
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Slide 9-14