Angles of Elevation and Depression

Angle of Elevation

When you see an object above you

When you see an object above you Angle of Elevation line of sight horizontal You There is an angle of elevation between the horizontal and your line of sight to the object

When you see an object above you Angle of Elevation line of sight horizontal You You can use Trigonometry to find distances from the Angle of Elevation Information

Example

Angle of Elevation 20º Door ramp 4 ft horizontal ground level A homeowner is to construct a ramp to his front door to make it wheelchair-accessible. How long is the ramp if the door is 4 ft above ground level and the angle of elevation is 200 Door ramp Angle of Elevation 4 ft 20º Bottom of Ramp horizontal ground level

Let x = the length of the ramp A homeowner is to construct a ramp to his front door to make it wheelchair-accessible. How long is the ramp if the door is 4 ft above ground level and the angle of elevation is 200 x 4 ft 20º Let x = the length of the ramp

The length of the ramp should be 11.7 ft A homeowner is to construct a ramp to his front door to make it wheelchair-accessible. How long is the ramp if the door is 4 ft above ground level and the angle of elevation is 200 The length of the ramp should be 11.7 ft x 4 ft 20º

Another Example

One is at a 62º angle of elevation A bystander notices two window washers on the side of a building, one straight up from the other. One is at a 62º angle of elevation And the other at a 30º angle of elevation. If the bystander is 42 ft from the point at the base of the wall directly below the window washers, How far apart are the window washers?

From 42 ft from the base of the wall 62º 42 62º angle of elevation From 42 ft from the base of the wall A bystander notices two window washers on the side of a building, one straight up from the other. One is at a 62º angle of elevation and the other at a 30º angle of elevation. If the bystander is 42 ft from the point at the base of the wall directly below the window washers, How far apart are the window washers?

From 42 ft from the base of the wall 30º 42 30º angle of elevation From 42 ft from the base of the wall A bystander notices two window washers on the side of a building, one straight up from the other. One is at a 62º angle of elevation and the other at a 30º angle of elevation. If the bystander is 42 ft from the point at the base of the wall directly below the window washers, How far apart are the window washers?

x y 62º 30º 42 Let x = distance from the ground of the highest window washer Let y = distance from the ground of the lowest window washer Then x – y = distance between the window washers

x 62º 42 Let x = distance from the ground of the highest window washer

x 62º 42 Let x =

x y 62º 30º 42 Let x = Let y = distance from the ground of the lowest window washer

y 30º 42 Let y = distance from the ground of the lowest window washer

x y 62º 30º 42 Let x = Let y =

x y 62º 30º 42 Let x = Let y = Then x – y = distance between the window washers

The Window Washers are 54.7 ft apart x y 62º 30º 42 The Window Washers are 54.7 ft apart

You Try

The world’s tallest fountain is in Fountain Hill, Arizona The world’s tallest fountain is in Fountain Hill, Arizona. If the angle of elevation to the top of the fountain from a point 755 ft from its base is 36.50 , find the height of the fountain Solution Next Slide

The height of the fountain is 558.7 ft The world’s tallest fountain is in Fountain Hill, Arizona. If the angle of elevation to the top of the fountain from a point 755 ft from its base is 36.50 , find the height of the fountain x 36.5º 755 The height of the fountain is 558.7 ft

Angle of Depression

When you see an object below you

When you see an object below you horizontal Angle of Depression line of sight Object There is an angle of depression between the horizontal and your line of sight to the object

We make use of the angle of elevation between the the two objects When given an angle of depression between the horizontal and your line of sight to the object. We make use of the angle of elevation between the the two objects To Solve Problems

Angle of Elevation Angle of Depression You horizontal line of sight Object The angle of depression between the horizontal and your line of sight to the object is equal to the angle of elevation between the the two objects

The angle of depression and the angle of elevation are alternate interior angles and therefore equal

Angle of Elevation Angle of Depression You horizontal line of sight Object The angle of depression between the horizontal and your line of sight to the object is equal to the angle of elevation between the the two objects

Example

A building casts a shadow 42 m long A building casts a shadow 42 m long. The angle of depression from the top of the building to the tip of the shadow is 53°. Determine the height of the building. Draw a diagram. Shadow 42 m

A building casts a shadow 42 m long A building casts a shadow 42 m long. The angle of depression from the top of the building to the tip of the shadow is 53°. Determine the height of the building. Draw a diagram. 53º 42 m

Let x = the height of the building A building casts a shadow 42 m long. The angle of depression from the top of the building to the tip of the shadow is 53°. Determine the height of the building. Draw a diagram. 53º x 53º 42 m Let x = the height of the building

The height of the building is 55.7 m A building casts a shadow 42 m long. The angle of depression from the top of the building to the tip of the shadow is 53°. Determine the height of the building. Draw a diagram. x 53º 42 m The height of the building is 55.7 m

Another Example

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram.

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 150

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 220 150

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 220 Angle of Depression 150

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 220 Angle of Depression 370 150

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. Angle of Depression 370

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 14º Angle of Depression 370

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 14º 370 14º

14º 370 x Let x = the distance from the ship to the base of the cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 370 14º x Let x = the distance from the ship to the base of the cliff

A lighthouse 220 ft tall sits on the edge of a 150 ft cliff A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 370 14º x

The distance from the ship to the base of the cliff is 1,484 ft A lighthouse 220 ft tall sits on the edge of a 150 ft cliff. The angle of depression from the top of the lighthouse to a ship at sea is 14°. Determine the distance from the ship to the base of the cliff. Draw a diagram. 370 14º x The distance from the ship to the base of the cliff is 1,484 ft

You Try

The angle of depression of an airplane to the airport is 30 The angle of depression of an airplane to the airport is 30.1 when the plane is 10.3 km from the airport measured along the ground. Determine the altimeter reading inside the plane or in other words the height of the plane from the ground. (Make a sketch, label all given angles and dimensions – Round to the nearest tenth) Solution Next Slide

The Altimeter reading is 5.97 km The angle of depression of an airplane to the airport is 30.1 when the plane is 10.3 km from the airport measured along the ground. Determine the altimeter reading inside the plane or in other words the height of the plane from the ground. (Make a sketch, label all given angles and dimensions – Round to the nearest hundredth) 30.1º x 30.1º 10.3 km The Altimeter reading is 5.97 km