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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Trigonometry Exploration 1 (cont’d)
Use the the triangles provided for you and the two new ones that you created to fill out the charts
below. You will need to use a protractor to measure the angles.
How do the side lengths of triangles DEF and GHI compare to those of triangle ABC?
How do the angles of triangles DEF and GHI compare to those of triangle ABC?
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Chart 2
Calculate the following:
Ratio
Fraction
Decimal
Fraction
Decimal
What do you notice about the results
found in these charts?
a/c
b/c
a/b
Ratio
How would you explain what you
have found to a classmate who was
absent today?
d/f
e/f
d/e
Ratio
Fraction
Decimal
g/i
Compare your answers with a
classmate. Are there any differences?
Why?
h/i
g/h
If I created another triangle and labelled it JKL (see diagram below) and this triangle was a different size
but exactly the same shape as ABC, what would the following ratios be?
j/l =
k/l =
j/k =
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Trigonometry Exploration 2
Using a ruler and a protractor and the grid on the reverse, create a triangle that has the following
characteristics:
(You will have to measure, you will not be able to “count blocks”.)
Angle A = ___________
Angle B = ___________
Side b = ____________
Fill out the missing information once you have finished carefully drawing the triangle. You will need to
carefully measure the angles and sides. Record your answers to the nearest millimetre.
Side a =
Side b =
Side c =
Complete the ratios. Write your answer as a fraction, then also as a decimal.
a/c =
b/c =
a/b =
Now find another person who had the same triangle. It will have exactly the same shape, but
might be a different size.
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Foundations
oundations of Math 10
Chapter 3 – Right Triangle Trigonometry
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Now with your triangle partners, compare the ratios you calculated. Come to some agreement for the
first 3 decimal values for each ratio. Record your answers below.
a/c =
b/c =
a/b =
Together with your partners, answer the following questions:
1. What determines the shape of a triangle? Explain your answer.
2. What have you noticed about the way the sides of identically-shaped triangles compare to oneanother even though they are different sizes?
3. Complete these sentences with the words in the table below. A word may be used more than
once or not at all.
Triangles that are exactly the same shape (have the _______________) but have different
______________ are called ______________ triangles. If two triangles are __________, then
the _____________ of their corresponding sides will be _____________.
In trigonometry, we can make use of these relationships to help us find missing information
about triangles, provided we have information about a triangle that is _______________ to the
one we are looking at.
similar
ratios
same angles
side lengths
equal
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Class Summary
As a class, we are going to fill out the table below. This will help us to find information about
triangles that are similar to the ones in the table.
Angle A
a/c
b/c
a/b
15 o
30 o
45 o
60 o
For example: Find the value of x.
18 cm
x
30°
7 cm
60°
x
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
DAY 3 – SINE, COSINE, AND TANGENT
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Foundations
oundations of Math 10
Chapter 3 – Right Triangle Trigonometry
We can use what we have found out about triangles to solve many problems. It is an entirely new area
of mathematics to you and it has different rules and operations than what you are used to with
arithmetic and algebra. In the past few lessons we learned that triangles that are exactly the same
shape (same angles) but have different sizes are called similar triangles.
triangles. If two triangles are similar then
the ratios of their corresponding sides will be equal. In special
special cases, when one angle of the triangle is a
right angle (90 ) then the triangle is called a right triangle.
triangle
We can use ratios of similar triangles to find missing lengths or angles, just as we did in the beginning of
the activity today. What happens if I wanted to find information about a triangle with an angle of 11 ,
or 23 , or some other angle? It would be very time consuming to draw and measure each triangle and
then calculate the missing side or angle. In the
the past there used to be very long trig tables (like we made,
but longer) and you could look up the ratios. Let’s take another look……
If we look at the labelled triangle we can see that a is
A
opposite angle A, and that b is adjacent (next) to angle
b
A and c is the longest side (across from the right angle)
c
and it is called the hypotenuse.
C
B
a
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Foundations
oundations of Math 10
Chapter 3 – Right Triangle Trigonometry
This means if we look from angle A, the ratios can be expressed as:
This is referred to as the sine of angle A, or
.
This means that if I had a triangle with an angle of 40◦
40 and I wanted to find out what the ratio of the
opposite side to the hypotenuse was (or
) I would look up or calculate
This is referred to as the cosine of angle A, or
.
This is referred to as the tangent of angle A, or
.
.
Let’s take another look at your class trig table. Use your calculator to find each ratio. Does it match
your table from the last lesson?
Angle A
a/c
b/c
a/b
15
30
45
60
90
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Foundations
oundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Now we can use the sine, cosine, and tangent ratios to find missing sides and angles in right triangles.
Ex.. Find the measure of x.
Look at the angle. What sides do we have?
56 cm
______________ and ______________. Which
trigonometric ratio is this? Set up the ratio.
x
33°
Solve for x..
Ex. Find
.
A
35
B
16
23
C
Ex.. Find the measure of t.
U
23◦
t=
S
s = 15 m
T
Assignment: With your partner, do the questions starting on page 120 #1,2, 4, 6bd, 8.
Assignment:
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
DAY 4 – SOLVING RIGHT TRIANGLES (I)
To “solve” a triangle means to find all missing sides and all missing angles. We have learned how to find
missing sides using the appropriate ratio, but there are some other methods for solving triangles that we
can use.
Ex. Find the missing side.
16 cm
20 cm
Ex. Find the indicated angle.
Ө
17 m
11 m
If we look from the angle we see we have two sides;
the ____________________ and the
______________________. We know that this
represents the ________________ ratio. So we can
set up the equation:
If we had that ratio in our table we could look up the angle (we could use the big table we made –
try that first). In the past, we would have had a book that we could look up the ratios in. But today
we will use a calculator. We already know the ratio, but we need the angle so we have to use a
reverse function, or inverse function.
Find the inverse function for the ___________ ratio on your calculator. Enter the inverse function,
then the ratio and hit enter. The calculator “looks up” the angle that goes with that ratio and
shows the resulting angle on the screen. (Note: On some calculators you will have to type the
ratio first, then the inverse function.)
Show what you did here:
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Ex. Find the indicated angle.
21 in
15 in
Ө
Assignment: Do the questions starting on page 120 #3, 5, 6ac, 7, 12, 13 , and 16.
Activity: Design a poster/study sheet that shows all you know about right triangles so far.
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
DAY 5 – SOLVING RIGHT TRIANGLES (II)
To “solve” a triangle means to find all missing sides and all missing angles. We have explored a number
of methods to solve right triangles. What are these?
1. ____________________________________________
2. ____________________________________________
3. ____________________________________________
Using some or all of these methods, solve the triangles below.
A
Q
17 m
43 cm
11 m
C
B
25⁰
S
R
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
When looking up, the angle of elevation is
the angle your line of sight makes with the
horizontal.
angle of elevation
horizontal
angle of depression
When looking down, the angle of
depression is the angle your line of sight
makes with the horizontal.
Ex.
A guy wire runs from a point on the ground 5 metres from the base of a
flag pole to its top. The angle of elevation of the wire is 65°. How long is
the wire? Answer to the nearest tenth of a metre.
wire length, w
flag pole
65°
5m
ground
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Ex. A man standing on top of a building looks down to see the building’s shadow.
The angle of depression is 35⁰. What is the length of the building’s shadow if the
height of the building is 40 ft? Answer to the nearest tenth of a foot.
Ex. A farmer wants to determine the height of a tree he needs to chop down.
From a distance 10m from the tree he measures the angle of elevation to be 65⁰.
What is the height of the tree?
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Name: _________________________
Group Members: _______________________
Height of a Flagpole – Trigonometry Challenge
Part A:
Your challenge today is to use trigonometry to estimate the height of the flagpole
at the front of the school. You may use only the tools below (and a calculator).
(You may not climb the flagpole, nor any other part of the school). Show all your
methods and calculations below.
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Part B: Imagine someone has taken away your protractor. Can you think of a
way to find the height of the flagpole using only the metre stick, string and a tape
measure? Try it – explain your method and show all work below.
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Foundations of Math 10
Chapter 3 – Right Triangle Trigonometry
Answer the following questions in full sentences.
What strategy did your group employ to find the height?
Why did you decide to use this strategy?
Was the strategy effective? Is there anything that
would have improved it?
Describe in your own words what you learned about
trigonometry during this activity.
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