Download Solving Problems with more than one Right Triangle

LESSON 6: SOLVING PROBLEMS INVOLVING MORE THAN ONE
RIGHT TRIANGLE
Learning Outcome: Learn to use trigonometry to solve problems modeled
by more than one right triangle.
Key Math Learnings: Many problems involve one or more right triangles
that may be in different planes.
The Muttart Conservatory in Edmonton has four climate-controlled square
pyramids, each representing a different climatic zone. Each of the tropical
and temperate pyramids is 24m high and the side length of its base is 26m.
Work with a partner.
Use this square based pyramid and label its height and base with the
measurements above.
Draw right triangles on the drawing that would help you determine the
angle between the edges of the pyramid at its apex.
How could you use trigonometry to help you determine this angle?
A
Need to use
Pythagorean theorem
to solve
24m
E
B
C
13m
D
𝐴𝐶 = √24² + 13² = 27.29
𝐼𝑛 ∆𝐴𝐶𝐸: tan 𝐴 =
13
13
= 𝑡𝑎𝑛−1 (
) = 25.46˚
27.29
27.29
𝐼𝑛 ∆𝐴𝐷𝐸: ∠𝐷𝐴𝐸 = 2(25.46˚) = 51˚
We can use trigonometry to solve problems that can be modeled using
right triangles. When more than one right triangle is involved, we have to
decide which triangles to start with.
Ex. Calculate the length of CD to the nearest tenth of a centimetre.
C
B
47˚
26˚
A
4.2cm
D
We need to use ∆𝐴𝐵𝐷 to calculate the length of BD.
Angle of elevation and angle of depression:
Both angles of elevation and angles of depression are always measured
from the horizontal. The angle of elevation looks from the horizontal
upwards:
Angle of elevation
And the angle of depression looks from the horizontal downwards:
Angle of depression
For example:
In Triangle ABC, the angle of elevation and the angle of depression are
indicated.
Angle of Depression
Angle of Elevation
Note: The angle of elevation and the angle of depression are always equal
Ex. From a height of 50m in his fire tower, a ranger observes the
beginnings of two fires. One fire is due west at an angle of depression of
9˚. The other fire is due east at an angle of depression of 7˚. What is the
distance between the two fires?
9˚
7˚
50 m
Need to solve for each triangle separately, and recognize that the angle of
elevation and depression are equal.
7˚
9˚
9˚
7˚
Ex. A surveyor stands at a window on the 9th floor of an office tower. He
uses a clinometers to measure the angles of elevation and depression of
the top and the base of a taller building. The surveyor sketches this plan of
his measurements. Determine the height of the taller building to the
nearest tenth of a metre.
31˚
42˚
39m
Assignment: pg. 18-121 #1-9, 11, 13-14, 18, 20