Transcript Chapter 8.4 and 8.5

CHAPTER 8.4
Trigonometry
Trigonometry
The word trigonometry comes from the Greek meaning “triangle
measurement”.
Trigonometry uses the fact that the side lengths of similar triangles are always
in the same ratio to find unknown sides and angles.
For example, when one of the angles in a right triangle is 30° the side opposite
this angle is always half the length of the hypotenuse.
8 cm
30°
12 cm
?
4 cm
?30°
6 cm
The sine ratio
The ratio of
the length of the opposite side
is the sine ratio.
the length of the hypotenuse
In trigonometry we use the Greek letter θ, theta, for the angle. The value of the sine
ratio depends on the size of the angles in the triangle.
O
P
P
O
S
I
T
E
H
Y
We say:
P
O
T
E
N
U
sin θ =
S
θ
E
opposite
hypotenuse
The sine ratio using a calculator
What is the value of sin 65°?
To find the value of sin 65° using a scientific calculator, start by making sure that your
calculator is set to work in degrees.
Key in:
sin
6
Your calculator should display 0.906307787
This is 0.906 to the nearest thousandth.
5
=
The cosine ratio
the length of the adjacent side
The ratio of
is the cosine ratio.
the length of the hypotenuse
The value of the cosine ratio depends on the size of the angles in the triangle.
H
Y
P
We say,
O
T
E
N
U
S
θ
ADJACENT
cos θ =
E
adjacent
hypotenuse
The cosine ratio
What is the value of cos 53°?
It doesn’t matter how big the triangle is because all right triangles with an angle of
53° are similar.
The length of the opposite side divided by the length of the hypotenuse will always
be the same value as long as the angle is the same.
In this triangle,
adjacent
cos 53° =
hypotenuse
10 cm
6
=
53°
6 cm
10
= 0.6
The cosine ratio using a calculator
What is the value of cos 30°?
To find the value of cos 30° using a scientific calculator, start by making sure that
your calculator is set to work in degrees.
Key in:
cos
3
0
Your calculator should display 0.866025403
This is 0.866 to the nearest thousandth.
=
The tangent ratio
The ratio of
the length of the opposite side
is the tangent ratio.
the length of the adjacent side
The value of the tangent ratio depends on the size of the angles in the triangle.
O
P
P
O
S
I
T
E
We say,
tan θ =
θ
ADJACENT
opposite
adjacent
The tangent ratio
What is the value of tan 71°?
It doesn’t matter how big the triangle is because all right triangles with an angle of
71° are similar.
The length of the opposite side divided by the length of the adjacent side will always
be the same value as long as the angle is the same.
In this triangle,
4 cm
opposite
tan 71° =
71°
adjacent
=
11.6
11.6 cm
4
= 2.9
Example 1
Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a fraction
and as a decimal to the nearest
hundredth.
Answer:
Example 1
Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a fraction
and as a decimal to the
nearest hundredth.
Answer:
Example 1
Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a fraction
and as a decimal to the
nearest hundredth.
Answer:
Example 1
Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a fraction
and as a decimal to the
nearest hundredth.
Answer:
Example 1
Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a fraction
and as a decimal to the nearest
hundredth.
Answer:
Example 1
Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a fraction
and as a decimal to the nearest
hundredth.
Answer:
Example 1
A. Find sin A.
A.
B.
C.
D.
Example 1
B. Find cos A.
A.
B.
C.
D.
Example 1
C. Find tan A.
A.
B.
C.
D.
Example 1
D. Find sin B.
A.
B.
C.
D.
Example 1
E. Find cos B.
A.
B.
C.
D.
Example 1
F. Find tan B.
A.
B.
C.
D.
Example 3
Estimate Measures Using
Trigonometry
EXERCISING A fitness trainer sets the incline on a
treadmill to 7°. The walking surface is 5 feet long.
Approximately how many inches did the trainer raise the
end of the treadmill from the floor?
Example 3
CONSTRUCTION The bottom of a handicap ramp is 15
feet from the entrance of a building. If the angle of the
ramp is about 4.8°, about how high does the ramp rise
off the ground to the nearest inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in.
Try it! Find each length using trigonometry.
• Pg 570 #3A and 3B
Example 4
Find Angle Measures Using Inverse Trigonometric
Ratios
Use a calculator to find the measure of P to the nearest
tenth.
Example 4
Use a calculator to find the measure of D to the nearest
tenth.
A. 44.1°
B. 48.3°
C. 55.4°
D. 57.2°
Try it! Find each angle measure.
• Pg. 571 #4A and 4B
Example 5
Solve a Right Triangle
Solve the right triangle. Round side measures to the
nearest hundredth and angle measures to the nearest
degree.
Example 5
Solve the right triangle. Round side measures to the
nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°,
AB = 13.6
B. mA = 54°, mB = 36°,
AB = 13.6
C. mA = 36°, mB = 54°,
AB = 16.3
D. mA = 54°, mB = 36°,
AB = 16.3
Example 1
Angle of Elevation
CIRCUS ACTS At the circus, a person in the audience at
ground level watches the high-wire routine. A 5-foot-6inch tall acrobat is standing on a platform that is 25 feet
off the ground. How far is the audience member from
the base of the platform, if the angle of elevation from
the audience member’s line of sight to the top of the
acrobat is 27°?
Make a drawing.
Example 2
Angle of Depression
DISTANCE Maria is at the top of a cliff and sees a seal in
the water. If the cliff is 40 feet above the water and the
angle of depression is 52°, what is the horizontal
distance from the seal to the cliff, to the nearest foot?
Make a sketch of the situation.
Since
are parallel,
mBAC = mACD by the
Alternate Interior Angles
Theorem.
Example 1
DIVING At a diving competition, a 6-foot-tall diver stands
atop the 32-foot platform. The front edge of the platform
projects 5 feet beyond the ends of the pool. The pool itself
is 50 feet in length. A camera is set up at the opposite end
of the pool even with the pool’s edge. If the camera is
angled so that its line of sight extends to the top of the
diver’s head, what is the camera’s angle of elevation to the
nearest degree?
A. 37°
B. 35°
C. 40°
D. 50°
Example 2
Luisa is in a hot air balloon 30 feet above the ground. She
sees the landing spot at an angle of depression of 34.
What is the horizontal distance between the hot air
balloon and the landing spot to the nearest foot?
A. 19 ft
B. 20 ft
C. 44 ft
D. 58 ft