Transcript Right Triangles

Page 269
Right Triangles
Objective
To solve right triangles.
To find the missing parts
of right triangles using
the trig functions.
Practical Application
The longest truck mounted ladder
used by the Dallas Fire Department
is 108 feet long and consists of four
hydraulic sections. Gerald Travis,
aerial expert with the department, indicates
that the optimum operating angle of this ladder
is 60°. Outriggers, with an 18-foot span between
each, are used to stabilize the ladder truck and
permit operating angles greater than 60°, allowing
the ladder truck to be closer to buildings in the
downtown streets of Dallas. Assuming the ladder
is mounted 8 feet off the ground, how far from an
84-foot burning building should the base of the
ladder be placed to achieve the optimum operating
angle of 60°? How far should the ladder be extended
to reach the roof?
Right triangles can be
used to define trig
functions.
B
c
a
tan A =
A
sin A =
cos A =
b
opposite
=
hypotenuse
C
a
c
adjacent
b
=
hypotenuse
c
opposite
a
=
adjacent
b
csc A = hypotenuse = c
opposite
a
sec A = hypotenuse = c
adjacent
b
b
cot A = adjacent =
a
opposite
SOH-CAH-TOA
This is a mnemonic device
commonly used for
remembering the first three equations
Opposite
Sin q =
Hypotenuse
Adjacent
Cos q =
Hypotenuse
Opposite
Tan q =
Adjacent
A right triangle has sides
whose lengths are 5 cm,
12 cm, and 13 cm. Find
the values of the six trig
functions of q.
sin q =
5
13
5
12
cos q =
13
q
12
5
tan q =
12
csc q =
13
5
13
sec q =
12
13
cot q =
12
5
Solve right triangle ABC. Round
angle measures to the nearest
degree and side measures to the
nearest tenth.
49° + B = 90°
Since A and B are complementary
B = 41°
sin 49° =
0.7547 ≈
c ≈ 9.3
7
c
7
c
tan 49° =
7
b
1.1504 ≈
7
b
A
49°
b ≈ 6.1
b
c
So, B = 41°, c = 9.3, and b = 6.1
41°
C
7
B
Find the measure of angle R to
the nearest degree.
sin R =
=
opposite
hypotenuse
8
14
Use your calculator
8 ÷ 14 = 2nd SIN = 34.849905
T
8
14
R ≈ 35°
S
R
The application at the beginning of the lesson.
Assume the ladder is mounted 8’ off the ground.
A. How far from the 84-foot burning building
should the base of the ladder be placed
to achieve the optimum operating
angle of 60°?
B. How far should the ladder be extended
to reach the roof?
tan 60° =
76
d
1.7321 ≈
76
d
d ≈ 43.9
sin 60° =
0.8660 ≈
l ≈ 87.8
76
l
78’
l
76
l
60°
d
So the truck should 43.9 feet from the wall and the ladder
should be extended 87.8 feet.
84’
Angles of Elevation
and Depression – page
272
• There are many other applications that require
trigonometric solutions. Surveyors use special
instruments to find the measures of angles of
elevation and angles of depression. An angle
of elevation is the angle between a horizontal
line and the line of sight from an observer to
an object at a higher level. An angle of
depression is the angle between a horizontal
line and the line of sight from the observer to
an object at a lower level.
A flagpole 40’ high stands on top of the Wentworth
Building. From a point P in front of Bailey’s drugstore,
the angle of elevation of the top of the pole is 54°54’,
and the angle of elevation of the bottom of the pole is
47°30’. To the nearest foot, what is the height of the
building?
Let x be the height of the building
and a be the distance from the
point to the foot of the building.
tan 47°30’ = x
tan 54°54’ = 40 + x
a
a
Solve each equation for a. Then the
following is true.
x
.
40 + x . =
tan 47°30’
tan 54°54’
40’
54°54’
x
tan 47°30’(40 +x) = x (tan 54°54’)
40tan 47°30’ = x(tan54°54’ – tan 47°30’)
43.6523 = 0.331x
x = 131.68 ≈ 132 feet
47°30’
a
Assignment
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