How High Is That Building?

Can anyone think of an object on the school campus that we could not use a tape measure to directly measure?

Does anyone have any idea how we can find the height of an object we can’t directly measure?

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If we use a right triangle to find the height of the object, which side of the triangle can we directly measure?

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In our real life example, what distance is represented by the base of the triangle?

In our real life example, what distance is represented by the base of the triangle?

The distance from where we are standing to the building

d

What kind of instrument can we use to measure the acute angle at the base of the triangle?

What kind of instrument can we use to measure the acute angle at the base of the triangle?

Which trigonometric function can we use to find the length of the opposite side if we know the length of the side adjacent to the acute base angle and the measure of the angle?

Known angle

Adjacent Side

Opposite SideHypotenuse

adjacentoppositehypotenuseadjacenthypotenuseopposite

tan

cos

sin

Which trigonometric function can we use if we know the length of the side adjacent to the acute base angle and the measure of the angle?

Known angle

Adjacent Side

Opposite Side

Hypotenuseadjacentoppositetan

Find the height of the building if the distance from the building is 45 feet and angle of elevation is 27o .

Find the height of the building if the distance from the building is 45 feet and angle of elevation is 27o .

tani =dh

adjacentopposite

tan

Find the height of the building if the distance from the building is 45 feet and angle of elevation is 27o .

tani =dh

tan27o = 45'h

Find the height of the building if the distance from the building is 45 feet and angle of elevation is 27o .

tani =dh

tan27o = 45'h

45' * tan27o= h

Find the height of the building if the distance from the building is 45 feet and angle of elevation is 27o .

tani =dh

tan27o = 45'h

45' * tan27o= h22.9 feet= h

So the height of the building is 22.9 feet.

You want to drive your truck hauling an excavator under an overpass that has a clearance of 15.5 ft? You set up your surveying equipment and measure the distance to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the overpass?

You want to drive your truck hauling an excavator under an overpass that has a clearance of 15.5 ft? You set up your surveying equipment and measure the distance to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the overpass?

tani = distance to truckheight of load

height of load

Distance to truck

You want to drive your truck hauling an excavator under an overpass that has a clearance of 15.5 ft? You set up your surveying equipment and measure the distance to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the overpass?

tani = distance to truckheight of load

height of load

Distance to truck

tan19o = 48'

height of load

You want to drive your truck hauling an excavator under an overpass that has a clearance of 15.5 ft?. You set up your surveying equipment and measure the distance to the load to be 48 feet and an angle of elevation 19°. Will your load fit under the overpass?

tani = distance to truckheight of load

height of load

Distance to truck

tan19o = 48'

height of load

48' * tan19o = height of load

The excavator will not fit under the overpass.

16.5' = height of load

You can walk across the Sydney Harbor Bridge and take a photo of the Opera House from about the same height as top of the highest sail.

This photo was taken from a point about 500 m horizontally from the Opera House and we observe the waterline below the highest sail as having an angle of depression of 8°. How high above sea level is the highest sail of the Opera House?

tan 8° = h/500 h = 500 tan 8° = 70.27 m. So the height of the tallest point is around 70 m.

A tree casts a shadow 70 feet long at an angle of elevation of 30º. How tall is the tree?

A tree casts a shadow 70 feet long at an angle of elevation of 30º. How tall is the tree?

7030ºtan x

x 30ºtan70ftx 4.40

When building a cabin with a 4/12 pitch roof, at what angle should the plumb cut of the rafter be cut?

When building a cabin with a 4/12 pitch roof, at what angle should the plumb cut of the rafter be cut?

o

adjacentopposite

6.714

12tan

412tan

tan

1

So the plumb cut will be o6.71

A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°. How tall is the flagpole?

A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°. How tall is the flagpole?

a = 50 tan 48° » 55.5 ft.

oa 48tan50

Let a denote the height of the flagpole.

Two trees stand opposite one another, at points A and B, on opposite banks of a river.

Distance AC along one bank is perpendicular to BA, and is measured to be 100 feet. Angle ACB is measured to be 79°. How far apart are the trees; that is, what is the width w of the river?

Two trees stand opposite one another, at points A and B, on opposite banks of a river.

Distance AC along one bank is perpendicular to BA, and is measured to be 100 feet. Angle ACB is measured to be 79°. How far apart are the trees; that is, what is the width w of the river?

 w 100 = tan 79° ,

w = 100 × tan 79° = 514.5 ft

Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. How tall is the building. How tall is the antenna itself, not including the height of the building?

Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits on the front edge of the roof of the building. The angle to the top of the antenna is 52°. How tall is the building. How tall is the antenna itself, not including the height of the building?

Let a represent the height of the building and h the height of the antenna.  Then the following relationships hold:

a = 50 tan 40°, a + h = 50 tan 52°

50 tan 40° + h = 50 tan 52°

h = 50 ( tan 52° - tan 40° ) » 22 ft.

You are looking up at a fourth story window, 40 feet up in a building. You are 100 feet away from the building, across the street. What is the angle of elevation?

You are looking up at a fourth story window, 40 feet up in a building. You are 100 feet away from the building, across the street. What is the angle of elevation?

The building is perpendicular to the ground. Therefore the 40 feet opposite the angle of elevation A and the 100 feet you are away from the building gives us

0.4 gives us angle A is approximately 21.8º

tani =dh

tani = 10040

tani = .4tan- 1(.4) = ii = 21.8o

Now, let’s get outside and try it !