7.5 Angles of Elevation and Depression
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7.5 Angles of Elevation and Depression. Objectives. Solve problems using angles of elevation Solve problems using angles of depression. Angles of Elevation. An angle of elevation is the angle between the line of sight and the horizontal when an observer looks upward. .

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7.5 Angles of Elevation and Depression

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7 5 angles of elevation and depression

7.5 Angles of Elevation and Depression


Objectives

Objectives

  • Solve problems using angles of elevation

  • Solve problems using angles of depression


Angles of elevation

Angles of Elevation

  • An angle of elevation is the angle between the line of sight and the horizontal when an observer looks upward.

The angle of elevationis always measured from the ground up.  It is always INSIDE the triangle.

In the diagram above, x marks the angle of elevation to the top of the tree as seen from a point on the ground. You can think of the angle of elevation in relation to the movement of your eyes.  If you are looking straight ahead, then you must raise (elevate) your eyes to see the top of the tree.


Example 1

CIRCUS ACTS At the circus, a person in the audience 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

Example 1:

Make a drawing.


Example 11

Divide both sides by tan

Example 1:

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.

Simplify.


Example 12

Example 1:

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


Your turn

Answer: about

Your Turn:

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?


Angles of depression

Angles of Depression

  • An angle of depression is the angle between the line of sight when an observer looks downward and the horizontal.

The angle of depressionis always OUTSIDE the triangle.  It is never inside the triangle.

In the diagram above, x marks the angle of depression of a boat at sea from the top of a lighthouse.

Again, you can think of the angle of depression in relation to the movement of your eyes.  If you are standing at the top of the lighthouse and you are looking straight ahead, then you must lower (depress) your eyes to see the boat in the water.


Relationship between angles of elevation and depression

Relationship between Angles of Elevation and Depression

  • As seen in the diagram, the dark black horizontal line is parallel to side CA of triangle ABC.  This forms two alternate interior angles which are equal in measure.  Thus,

the angle of elevation = the angle of depression


Example 2

SHORT-RESPONSE TEST ITEM A wheelchair ramp is 3 meters long and inclines at Find the height of the ramp to the nearest tenth centimeter.

Read the Test ItemThe angle of depression between the ramp and the horizontal is Use trigonometry to find the height of the ramp.

Solve the Test ItemMethod 1The ground and the horizontal level with the platform to which the ramp extends are parallel. Therefore, since they are alternate interior angles.

Example 2:


Example 21

Y

W

Answer: The height of the ramp is about 0.314 meters,

Example 2:

Mulitply each side by 3.

Simplify.


Example 22

Method 2The horizontal line from the top of the platform to which the wheelchair ramp extends and the segment from the ground to the platform are perpendicular. So, and are complementary angles. Therefore,

Y

W

Example 2:


Example 23

Answer: The height of the ramp is about 0.314 meters,

Example 2:

Multiply each side by 3.

Simplify.


Your turn1

SHORT-RESPONSE TEST ITEM A roller coaster car is at one of its highest points. It drops at a angle for 320 feet. How high was the roller coaster car to the nearest foot before it began its fall?

Your Turn:

Answer: The roller coaster car was about 285 feet above the ground.


Example 3

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 Find the distance between the two dolphins to the nearest meter.

Example 3:


Example 31

are right triangles. The distance between the dolphins is JK or Use the right triangles to find these two lengths.

Because are horizontal lines, they are parallel. Thus, and because they are alternate interior angles. This means that

Example 3:


Example 32

Divide each side by tan

Example 3:

Multiply each side by JL.

Use a calculator.


Example 33

Divide each side by tan

Answer: The distance between the dolphins is , or about 8 meters.

Example 3:

Multiply each side by KL.

Use a calculator.


Your turn2

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 Find the distance between the two cars.

Your Turn:

Answer: about 24 feet


Assignment

Assignment

  • Pre-AP Geometry:Pg. 373 #4 – 16, 20, 26

  • Geometry:Pg. 373 #4 - 16


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