Transcript Angle of Depression

LESSON 8–5
Angles of Elevation and
Depression
Five-Minute Check (over Lesson 8–4)
TEKS
Then/Now
New Vocabulary
Example 1: Angle of Elevation
Example 2: Angle of Depression
Example 3: Use Two Angles of Elevation or Depression
Over Lesson 8–4
Use a calculator to find tan 54°.
A. 0.6738
B. 0.7432
C. 0.8090
D. 1.3764
Over Lesson 8–4
Find mB to the nearest tenth of a degree if
cos B = 0.8926 and B is an acute angle.
A. 63.2
B. 26.8
C. 18.4
D. 0.999
Over Lesson 8–4
Find x. Round to the
nearest tenth.
A. 14.1
B. 17.4
C. 19.4
D. 21.3
Over Lesson 8–4
Find x. Round to the
nearest tenth.
A. 9.5
B. 15.9
C. 23.7
D. 30.8
Over Lesson 8–4
Find x. Round to the
nearest tenth.
A. 34.7
B. 43.8
C. 46.2
D. 52.5
Over Lesson 8–4
Use a special right triangle to express sin 45° as
a fraction.
A.
B.
C.
D.
Targeted TEKS
G.9(A) Determine the lengths of sides and measures of angles
in a right triangle by applying the trigonometric ratios sine,
cosine, and tangent to solve problems.
Mathematical Processes
G.1(A), G.1(B)
You used similar triangles to measure
distances indirectly.
• Solve problems involving angles of elevation
and depression.
• Use angles of elevation and depression to
find the distance between two objects.
• angle of elevation
• angle of depression
Angle of Elevation
CIRCUS ACTS At the circus, a person in the
audience at ground level watches the high-wire
routine. A 5-foot-6-inch 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.
Angle of Elevation
Since QR is 25 feet and RS is 5 feet 6 inches or 5.5 feet,
QS is 30.5 feet. Let x represent PQ.
Multiply both sides by x.
Divide both sides by tan
Simplify.
Angle of Elevation
Answer: The audience member is about 60 feet from
the base of the platform.
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°
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.
Angle of Depression
Let x represent the horizontal distance from the seal to
the cliff, DC.
C = 52°; AD = 40, and
DC = x
Multiply each side by x.
Angle of Depression
Divide each side by tan 52°.
Answer: The seal is about 31 feet from the cliff.
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
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.
Use Two Angles of Elevation or Depression
Analyze
ΔMLK and ΔMLJ are right triangles.
The distance between the dolphins is
JK or JL – KL. Use the right triangles
to find these two lengths.
Formulate
Because
are horizontal
lines, they are parallel. Thus,
and
because they are alternate interior
angles. This means that
Use Two Angles of Elevation or Depression
Determine
Multiply each side by JL.
Divide each side by tan
Use a calculator.
Use Two Angles of Elevation or Depression
Multiply each side by KL.
Divide each side by tan
Use a calculator.
Answer: The distance between the dolphins is
JL – KL. JL – KL ≈ 219.93 – 211.96,
or about 8 meters.
Use Two Angles of Elevation or Depression
Justify Substitute the value for JK and KL in for the
equation
(JK + KL)tan35° = 154
(8 + 211.96 )tan35° = 154
154 = 154
Evaluate When using the angles of depression to two
different objects to calculate the distance between
them, it is important to remember that the two
objects must lie in the same horizontal plane. Eight
meters is reasonable for the distance between the
dolphins.
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
LESSON 8–5
Angles of Elevation and
Depression