Transcript Chapter 8.4 and 8.5

CHAPTER 8.4 AND 8.5
Trigonometry and Angles of
Elevation and Depression
Trigonometry
• Trigonometry is the study of triangle measurement
• A trigonometric ratio is a ratio of the lengths of two
sides of a right triangles.
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 2
Use Special Right Triangles to Find Trigonometric
Ratios
Use a special right triangle to express the cosine of 60° as
a fraction and as a decimal to the nearest hundredth.
Draw and label the side lengths of a
30°-60°-90° right triangle, with x as
the length of the shorter leg and 2x
as the length of the hypotenuse.
The side adjacent to the 60° angle
has a measure of x.
Example 2
Use a special right triangle to express the tangent of 60°
as a fraction and as a decimal to the nearest hundredth.
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?
Let y be the height of the treadmill from the floor in
inches. The length of the treadmill is 5 feet, or 60 inches.
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
Angles of Elevation and Depression
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
Example 3
Use Two Angles of Elevation or Depression
DISTANCE Vernon is on the top deck of a cruise ship and
observes two dolphins following each other directly
away from the ship in a straight line. Vernon’s position
is 154 meters above sea level, and the angles of
depression to the two dolphins are 35° and 36°. Find the
distance between the two dolphins to the nearest
meter.
Example 3
Madison looks out her second-floor window, which is 15
feet above the ground. She observes two parked cars.
One car is parked along the curb directly in front of her
window and the other car is parked directly across the
street from the first car. The angles of depression of
Madison’s line of sight to the cars are 17° and 31°. Find
the distance between the two cars to the nearest foot.
A. 14 ft
B. 24 ft
C. 37 ft
D. 49 ft