SECTION 8.5

ANGLES OF ELEVATION AND DEPRESSION

An angle of elevation is the angle formed by a horizontal line and observer’s line of sight to an object above the horizontal line.

An angle of depression is the angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line.

Since horizontal lines are parallel, the angle of elevation and the angle of depression are congruent by the alternate interior angles theorem.

Example 1:

a) 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°?

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.

oppositetan 27 tan

adjacent

30.5tan 27 30.5,

tan 27 30.5 Multiply both sides by .

30.5Divide both sides by tan 27 .

tan 2759.9 Simplify.

QS

PQ

QS PQ xx

x x

x

x

The audience member is about 60 feet from the base of the platform.

Example 1:

b) 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?

40°

Example 2: 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?

Let x represent the horizontal distance from the seal to the cliff, DC.

opptan tan

adj

40tan52 52 ; 40,and

tan52 40 Multiply each side by .

40Divide each side by tan 52 .

tan5231.3

ADC

DC

C AD DC xx

x x

x

x

The seal is about 31 feet from the cliff.

Example 3:

a) 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.

154 oppositetan35 tan ; 35

adjacent

tan35 154 Multiply each side by .

154Divide each side by tan35 .

tan35219.93 Use a calculator.

m MJLJL

JL JL

JL

JL

154 oppositetan36 tan ; 36

adjacent

tan36 154 Multiply each side by K .

154Divide each side by tan36 .

tan36211.96 Use a calculator.

m MKLKL

KL L

KL

KL

The distance between the dolphins is JK – KL. JL – KL ≈ 219.93 – 211.96, or about 8 meters.

Example 3:

b) 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.

24 feet