MATH on the Job
Judith Dupuis uses her math skills to calculate how long it will take her plane to ascend or descend safely.
Judith Dupuis is a pilot. She earned her Commercial Pilot-Airline Transport Pilot Licence and her Class 2 Instructor Rating. She instructs new pilots at the Springbank Airport in Springbank, Alberta. She also works with the Mission Aviation Fellowship of Canada and will be going to Africa to train pilots and fly cargo, food, and relief supplies to people in remote locations.
She will also help to transport people in those areas.
When landing her plane, Judith has to approach the runway at a precise angle of descent, or glide slope. Modern airports have visual guidance systems that allow pilots to see whether they are on the correct approach angle. At an elevation of 344 ft, Judith sees one white light and one red light on the runway Visual Approach Slope Indicator (VASI). This tells her that her angle is correct, at 3°. How far is Judith from her touchdown point?
Judith observes that her bearing is 42°. How far south and west is she from her landing point?
explore the Math
Trigonometry is a very useful tool. As you saw in the previous section, it can be used to calculate lengths, distances, and angles.
These values may be useful in themselves, or you can use them to calculate other quantities, such as volume and surface area. The information from one right triangle can also be used to solve any other right triangle with which it shares a common edge. In the last section, this idea was used to solve 4 consecutive triangles on a kite, each with less information given than the previous one.
this airplane is a de havilland beaver, commonly known as a bush plane. the beaver is a single-engine plane that was built in Canada.
Because the real world is three-dimensional, you cannot draw every situation on a flat piece of paper. You can still use the trigonometric principles, but the separate triangles may exist at an angle to each other, and only share common edges. For example, Judith’s flight path can be drawn as two triangles, one on the ground (using her
south and west distances) and one standing up vertically, using her elevation.
Make a sketch of Judith’s distance calculation problem, showing the two triangles, and identify the edge shared by the two. Which triangle must you solve first?
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example 1
before development takes place, surveyors determine the elevation of any inclines on the land.
A tunnel through a hill has an angle of elevation of 3.5°, and a direction of 22°
north of east. The length of the tunnel along its floor is 1400 m. Examine the diagram below.
1400 m h
∆East ∆North
∆Elevation N Entrance 3.5° Exit
22°
A B
C
a) What is the difference in elevation between the entrance and the exit of the tunnel?
b) How far north of the entrance is the tunnel exit? How far east is the exit?
solutIon
a) To find the difference in elevation, consider the triangle ABC. You know the length of the hypotenuse and the angle of elevation. You need to calculate the height of the side opposite the angle of elevation—the change in elevation—so that you can use the sine ratio.
sin θ = opp hyp sin 3.5° = elevation
1400 1400 sin 3.5° = elevation
85 ≈ elevation The change in elevation is 85 m.
b) To calculate the distances north and east of the tunnel entrance, you must first calculate the horizontal distance between the two ends.
adj
hyp = cosθ 1400h = cos 3.5°
h = 1400 cos 3.5°
h ≈ 1397 m
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∆North
1397 = sin 22°
∆North = 1397 sin 22°
∆North ≈ 523 m
∆East
1397 = cos 22°
∆East = 1397 cos 22°
∆East ≈ 1295 m
The tunnel exit is 523 m north and 1295 m east of the entrance.
example 2
Justin is laying out a ramp to go up a 30° slope in a park. Because the ramp may not have a slope greater than 1:20 so that it can be used comfortably by wheelchairs and motorized scooters, he must angle it across the slope. The total change in elevation is 4 ft.
C
b 30°
A 4 ft w
1:20 ℓ
a) What is the angle that the ramp must make with the base of the hill?
b) What width of hill is needed to build the ramp without using a switchback?
c) How can Justin use the calculated information to lay out the ramp without having to measure an angle on the slope?
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solutIon
a) To calculate the angle, Justin needs two pieces of information, for example, ℓ and b.
Angle C can be calculated knowing that the ramp has a slope of 1:20.
As we saw in chapter 1, this ratio represents the tangent of an angle having a rise of 1 units and a run of 20 units.
tan C = rise run tan C = 1
20 tan C = 0.05
C = tan−1(0.05) C ≈ 2.862°
sin C = change in elevation
sin C = 4
= 4 sin C
= 4
sin 2..862
≈ 80.111 ft
sin 30° = change in elevation b
sin 30° = 4 b
b = 4
sin 30°
b = 8 ft
The angle A can now be calculated using lengths b and ℓ:
sin A = b
sin A = 8
80.111 sin A = 0.100
A = 5.731°
189 Chapter 4 Trigonometry of Right Triangles
these people are constructing a concrete ramp.
b) Justin can calculate the width using the cosine ratio.
cos A = adjacent hypotenuse cos A = w
w = cos A
w = 80.111cos 5.731 w ≈ 79.711
The width is 79.711 feet, rounded to 80 feet.
alternatIve solutIon
Using the values for ℓ and b calculated in part a), Justin can find the width using the Pythagorean theorem.
w2 + b2 = 2 w2 = 2 − b2 w2 = 80.1112 − 82 w2 = 6417.77 − 64
w = 6353.77 w ≈ 79.711
c) To lay out the ramp, Justin can measure 79.711 ft from the starting point along the base of the hill, and then straight up the hill to the next level to get the end point of the ramp.
Mental Math and estimation
The sine of 36° is equal to 0.5878 (to four-decimal accuracy). Without using a calculator, show how you can determine the cosine of 54°, to four-decimal accuracy.
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dIsCuss the Ideas
usIng trIgonoMetry to CalCulate heIght
Arborists can use trigonometry to calculate the height of a tree.
An arborist needs to measure the height of a tree. It isn’t practical to use a tape measure, but she knows that she can use trigonometry to calculate the height.
1. How much information does she need to calculate the height of a tree?
2. Which pieces of information would be easiest for her to measure in order to use trigonometry?
The angle between a horizontal line of sight and the line of sight to an object that is higher than the observer is called the angle of elevation. If the object is lower than the observer, the angle between the horizontal line of sight and the line of sight to the object is called the angle of
depression. A simple device called a clinometer can be used to measure the angle of elevation or depression.
angle of elevation: the angle formed between the horizontal and the line of sight when looking upwards; sometimes referred to as the angle of inclination
horizontal angle of elevation line of sight
angle of depression:
the angle formed between the horizontal and the line of sight while looking downwards
horizontal angle of depression line o
f sight clinometer: an
instrument for measuring angles of elevation and depression
The arborist walks towards the tree until her clinometer indicates that the angle of elevation to the top of the tree is 45 degrees. She then measures the distance from that point to the tree, which is 23 metres.
3. Draw a sketch of the arborist’s position with respect to the tree and label it with the known information.
4. Which trigonometric ratio should she use to calculate the height of the tree?
5. Why does she use the 45-degree angle as her reference angle?
6. What is the height of the tree above her horizontal line of sight?
7. What measurement must she add to this height to get the true height of the tree?
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example 3
bylaws govern the height of buildings, such as these buildings in downtown Saskatoon, Saskatchewan.
Eloise is standing on the edge of a building. She can see that the angle of elevation to the top of the next building is 46° and the angle of depression to the bottom of the building is 62°. If the building is 30 m away, how tall is it?
solutIon
First, draw a diagram.
30 m x
y
46°
62°
Two tangent calculations are required. For both triangles, the angle and the adjacent side are known, and the opposite side is needed.
tan 46° = y 30
y = 30 tan 46°
y ≈ 31.1 m tan 62° = x
30
x = 30 tan 62°
x ≈ 56.4 m
To find the total height of the building, add x and y.
56.4 m + 31.1 m = 87.5 m
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example 4
Liu Hui was a third-century mathematician and author. This illustration demonstrates Hui’s method of calculating the height of an island while on the sea.
The Chinese mathematician, Liu Hui, wrote the Sea Island Mathematical Manual in the third century. In this manual, there was a method for sea captains to calculate the height of sea islands by measuring the angle sighted to the top of the island from two different points.
The angle sighted from the horizon to the top of a land feature or object is known as the angle of elevation.
Captain Ji sights an island in the distance. He uses a clinometer to measure the angle of elevation to the top of the island as 32°. He sails 400 m closer to the island and takes another sighting.
The angle of elevation to the top of the island from his closer position is 58°. What is the height of the island?
solutIon
The solution to this problem combines trigonometry and algebra. Since there are two unknowns, write two equations.
Captain Ji knows that he can solve this problem with trigonometry if he creates two imaginary right-angled triangles by drawing a line straight down from the peak of the island.
He can now write two tangent ratios.
193 Chapter 4 Trigonometry of Right Triangles
58° 32°
x y
400 m
tan 58° = opp adj tan 58° = y
x
tan 32° = opp adj tan 32° = y
x + 400
Liu hui is thought to have lived in what is now China’s Shanxi Province.
Rearrange the first equation.
y = x tan 58°
Substitute this in the second equation for y.
tan 32° = x tan 58°
x+ 400 (x + 400) tan 32° = x tan 58°
x tan 32° + 400 tan 32° = x tan 58°
400 tan 32° = x tan 58° − x tan 32°
400 ttan 32° = x(tan 58° − tan 32°) x = 400 tan 32°
tan 58° − tan 32°°
x ≈ 256.23 Substitute x ≈ 256.23 into y = x tan 58°.
y = 256.23 tan 58°
y ≈ 410.05 m The island is about 410 m high.
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ACtIVIty 4.3
fabrICate a range hood
Range hoods are installed above stoves to provide ventilation.
You have been asked to fabricate a range hood as shown in the diagram below.
750 mm
600 mm
100 mm 200 mm
400 mm 300 mm
c e
G h F
d
a b 40°
1. a) What is the slope of the side faces?
b) What are the angles at the bottom corner of each trapezoidal piece?
2. Draw a pattern for all of the pieces of the range hood and label them with their dimensions.
195 Chapter 4 Trigonometry of Right Triangles
3. Create a table similar to the one shown below and record the dimensions of your range hood.
range hood dIMensIons
Dimension Formula used Calculated value
a b c d e F G H
sa mp le
Next, draw each range hood piece on cardboard, cut out the pieces, and construct the hood.
Assemble your range hood pieces. How well the pieces fit together will depend on how accurately you measure the lengths and angles.
buIld your skIlls
this woman is using a transit.
1. Anna begins a geological survey at the peak of a mountain, which she knows is 8060 ft above sea level. She travels down the mountain and measures that she has travelled 2210 ft down the slope at an angle of depression of 15°. She then travels 2460 ft down the slope at an angle of depression of 23°, and a third leg of 890 ft down the slope at an angle of depression of 34°. What is the elevation above sea level at her final position?
23° 15°
34°
8060 ft 890 ft
2460 ft
2210 ft
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Rock climbers need agility and strength. before climbing, they may plot the course they will take to the top.
2. A climber, standing on the edge of Horsethief Canyon in Drumheller, Alberta, measures an angle of depression of 62° down to the bottom of the other side. From his position, at an angle of elevation of 14°, he notices a rock formation that reaches above the lip of the far side of the canyon.
He wants to climb from the bottom of the canyon to the top of the rock formation. What distance is that? If he wants to make sure that he has 10 ft of extra rope before attempting to climb it, how much rope does he need?
62°
14°
140 ft
3. For a research project, Jordan measures the depth of erosion of certain cliffs near Alberta’s Dry Island Buffalo Jump Provincial Park. In the past, Cree hunters stampeded animals over these cliffs as a hunting technique.
Eighty years ago, a surveyor recorded that the cliff was 49.07 m tall. Jordan stands 25 m away from the base of the cliff. The ground and the base of the cliff form a right angle. He finds that the angle of elevation is 62.9°.
How tall is the cliff now? How many centimetres has the cliff’s height lost to erosion? What was the angle of elevation 80 years ago?
over 100 species of birds, including golden eagles, have been observed in Alberta’s Dry Island buffalo Jump Provincial Park.
197 Chapter 4 Trigonometry of Right Triangles
4. Alan wants to drill a hole from point A to point B through a cedar block with 12-inch edge lengths, as shown in the diagram. At what horizontal angle a and vertical angle b must he drill to achieve this drill path?
b
Staff in an airport traffic control tower communicate with pilots to ensure that planes land or take off safely.
5. The straight-line distance of an airplane to a control tower is 36.2 km. If the angle of elevation from the tower to the plane is 4°, and the plane is 24 km south of the tower, how much west of the tower is the plane?
36.2 km
b
4°
24 km N
6. An architectural firm has been asked to design a hotel that will hang over the edge of a canyon and be built partially into a cliff. Before they design the building, they need to know the height of the cliff. It is difficult to measure the height of the cliff by physical means because of the irregular shape of the rock, and the bulk of the cliff is in the way to get a line of sight from point A. However, there is a line of sight from point B across the edge of the canyon, and the workers were able to make the following measurements.
Proposed hotel location bottom of the cliff?
b) What is the height of the cliff?
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7. Tim looks straight north and sees the top of a building 40 m distant at an angle of elevation of 20°. He looks straight west from the same position and sees the top of a building 80 m distant at an angle of elevation of 14°.
a) Sketch the situation.
b) Which building is taller, and by how much?
c) What is the straight-line distance between the top points of the buildings that Tim used to measure?
d) What is the angle of elevation from the top of the lower building to the top of the higher building?
extend your thinking
8. Marie-Claude is building an artist’s easel. She wants the painting to slope at an angle of 20° from the vertical. The angle at the peak between the front two legs is 30°, and all three legs are 2000 mm long.
a) Calculate the angle that the back leg will make with the ground when the easel is set up.
b) She wants to make sure that she sets the easel up at the same angle every time, so she connects the bottom of the back leg to the bottoms of each of the front legs with string. What length of string must she use to maintain the 20° angle of her painting?
A
ℓ6 ℓ5
ℓ1
ℓ3
ℓ2
ℓ4
30° 2000 m
2000 mm
70°