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TRANSCRIPT
© Community Learning Centre 2015 1
Detective Cosine McTrig’s Private Journal of
Mathematical Enlightenment
Grade 10 Mathematics Lesson: Trigonometry
Community Learning Centre 2015
© Community Learning Centre 2015 2
Curriculum Goals:
SOHCAHTOA Pythagorean Theorem Sum of angles in a
triangle Equilateral triangle
(angles, sides)
Making angles Problem Solving Critical Thinking Measurement Topography
Rate on a scale of one to ten your knowledge of how to:
Determine the relationship among linear scale factors, areas,
the surface areas and the volumes of similar figures and
objects.
Solve problems involving two right triangles.
Extend the concepts of sine and cosine for angle 0˚to 180˚.
Apply the Trigonometric Rules: sine, cosine and tangent to
solve problems
Solve problems involving distances between points in a coordinate plane.
Solve problems involving midpoint of line segments.
Determine the equation of line, given information that
uniquely determines the line.
Solve problems using slope of: parallel lines/ perpendicular lines.
How to find the square root of a number
How to calculate the height of an object using angle and
side
How to show real world measurements as ratios
© Community Learning Centre 2015 3
Table of Contents
Line Graphs 4
Area of a Triangle 6
Square Roots 9
Pythagorean Theorem 12
Rates, Ratios and Proportional Reasoning 14
Field Trip Problem Solving: 18
SOH CAH TOA 22
• Sine
• Cosine
• Tangent
© Community Learning Centre 2015 4
Part One: Line Graphs
Line graphs are used to chart information over a period of time or distance. Let’s look at the following graph:
1. If you measure the temperature of
chicken eggs, from the time they
are collected, to the time they
hatch you would record the
temperature over a period of time.
Look at the following data that was collected and chart it on the graph
below:
Day Temperature
Day 1 72˚
Day 5 85˚
Day 10 90˚
Day 15 95˚
Day 21 (hatch day)
100˚
Day 23 95˚
Day 35 80˚
© Community Learning Centre 2015 5
2. Match the stories to the graph:
a. Tessa started walking to school, then ran to get her
homework, and then ran the rest of the way to school.
b. Ben started his homework. He stopped for a snack and
then he continued with his homework.
3. As you are standing at the sink doing dishes one night, you
realize that you could probably make a lot of money doing
dishes. Face it: you are good at it. You imagine a graph in your head and start to count the dollars over time. If you charged
your mom 1 cent the first day and then twice as much as the
day before
the next
day, would you be rich
in 30 days? Hint: formula = 2 times the day as exponent, minus one
© Community Learning Centre 2015 6
Part Two: Area of a Triangle
1. How many squares are in this
rectangle:
2. How many squares are in this
triangle:
3. Find the area of the rectangles:
4. Find the area of the triangles:
5. Is there a formula you can use to simplify the work?
© Community Learning Centre 2015 7
6. Draw a dotted line to show the height of the triangle.
How long is the base?
How long is the height?
7. Draw three triangles. Measure the
angles for their sides using a protractor.
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8. Find the correct measurements of the following triangles
using a ruler and protector.
Base:
Height:
Area:
Base:
Height:
Area:
9. Draw a triangle with the following measurements:
Base: 6cm
Height: 2cm
© Community Learning Centre 2015 9
Part Three: Square Roots
1. What is the area of a triangle:
What is the area of the square in the centre:
2. Draw triangles on the blank square. What is the area of
the square:
© Community Learning Centre 2015 10
3. Let’s try and do the opposite now. What is the area of the triangle on the inside of the squares?
4. How do you find the square root of a triangle: add up all
the squares. Just like the roots of a tree; the square root
of a triangle is the area that it takes up. a. Add up the squares in a.
b. Add up the squares in b.
c. Add them together.
d. Divide the total by itself (to square it up again).
!! = !! = !! = !! + !!
© Community Learning Centre 2015 11
5. Something random by related…
a. Trace this shape onto paper or transparency.
b. Making one cut only, you should be able to make a perfect square with the two pieces you have
created!
c. Try it with the second shape!
© Community Learning Centre 2015 12
Part Four: Pythagorean Theorem
1. A shopping cart is rolling down a slope, heading for a car. Will it hit the car?
2. Highlight the side of the triangle that is “c” according to the Pythagorean Theorem.
3. What does n or x or y mean when it is using in math.
Why do we use letters instead of a question mark?
12
15 x
© Community Learning Centre 2015 13
4. Now that we know a squared + b squared gives us c
squared, solve the following:
a)
b)
c)
5. When people by a TV or computer, the size is determined
by the diameter of the screen. If there is a sale, such as
the following, which laptop gives you the best deal?
a. 13” TV = $399.00
b. 17” TV = $499.00
c. 21” TV = $599.00
X 6cm
3cm
r
14cm
20cm
F 5cm
3cm
© Community Learning Centre 2015 14
Ratios, Proportions & Units
Maps are great examples of real life ratios and proportions.
They were the first historical
tools that people
used to
communicate about where they
had been and
what they had
seen.
On this map, the
scale is 1 cm to
120 meter. This
is written as the following ratio:
1 : 250
The first number in the “scale
factor” is the
distance on the
map and the
second number is the distance in
real life.
© Community Learning Centre 2015 15
The scale factor is a ratio that shows the relationship between
the imaginary and the real. For example, 1:25,000 means that
for every 1cm on the map, 12,000cm are being represented in real life. In the case of our map, 1 cm on the map represents 1
meter in real life.
Try it out:
Measure the distance from the summit of Outlook Hill to the
point where Mad Brood meets Atwater Pond.
What is the distance in cm:
What is the distance in real life (m):
You can use these two numbers to create proportions (two equal ratios).
!!"!" !
= !"#$%&'( !"# !"#$%&"' !" !"# !" (!")!"#$%&'( !"# !"#$% !"#$ !" !"#$ !"#$ (!)
You can use cross-multiplying and dividing to find the missing
proportion.
Step 1: write the distance (D) on the map: _____ cm
Step 2: times D by the real life ratio (R) of 25m: ____m
Step 3: We usually use km rather than meters. In order to
convert to km, times R by 1000. (The ratio for meter
to kilometer is 1000:1)
© Community Learning Centre 2015 16
You want to go
camping at the sites on
the far side of Atwater Pond. One of the
people in your group
thinks that it is easier
to follow Randy’s River and Little Brook, and
another person in the
group thinks it is faster
to follow the road up to Clearbrook and
cross the mountain.
In order to solve the
growing dispute, you quickly measure each distance, you create a triangle on the
map connecting the bridge by the parking lot to the camp side
to Atwater lake along Randy’s River (A), Atwater lake to
Clearbrook(B), and Clearbook back to the bridge (C) using straight lines.
Next you write the length of each line in cm:
A = ______
B = ______ C = ______
You convert each distance to real-life (meters) using the scale
and times it by 1000 to represent km.
Which path is the fastest?
© Community Learning Centre 2015 17
When you get back to the car, someone realizes they left their
cell phone on the top of the mountain. You all agree that you
will go camp out at Randy’s River Campsite for the night, and then go the next morning to find the phone. As the night drags
on another argument breaks out. Some people think it is faster
to follow the river and then hike up the mountain from
MadBrook. The second group thinks it is faster to follow the road to Clearbrook and climb up from there.
Once again, you
show them, using Pythagorean
theorem, which
distance will take
the last amount of
time:
© Community Learning Centre 2015 18
Field Trip Problem Solving
1. Use the Pythagorean Theorem to find the
hypotenuse: a. .
b.
c.
n! = 4! + 3! = 16 + 9
= 25
n = 25
n = 5
n! = 3! + 6! =
=
n =
n =
n! = 2! + 7! =
=
n =
n =
4
3
6
3
2 7
© Community Learning Centre 2015 19
Using Pythagorean Theorem you can find any missing side!
2. To find the missing side using Pythagorean Theorem, use the same principle and estimate:
a. m
b.
c. Draw your own right
triangle and show how you solved to find the missing side.
n! = 7! + 3! = 49 + 9
= 58
n = 58
n = ~ 7.6
n! = 5! + 5! =
=
n =
n =
n! = ! + !
=
=
n =
n =
7
3
5
5
© Community Learning Centre 2015 20
3. One of the requirements to become registered as a
Big Tree is to know the height of the tree (shown above, the circumference of the trunk, and the canopy. Write down the two measurements you found for the canopy. Calculate the average canopy circumference using Pythagorean theorem:
© Community Learning Centre 2015 21
4. In order to measure the height of a tree you need two pieces of information: the distance from the tree to where you are standing and the angle from you to the top of the tree (A).
We measured the distance from the tree to where we could clearly see the top of the tree, which turned out to be 530 feet. We then used an angle meter to measure the angle from our eyes to the top of the tree. Note: after we find the height of the tree we will need to add our
own height, in feet, as well as any distance to the base of the tree if
it is below our feet level (as it was at Elkington Forest).
Unlike the previous questions which gave us side a and b so that we could find c, we only have one side and an angle. To find the answer, we will need to use Trigonometric Ratios: SOH CAH TOA!
© Community Learning Centre 2015 22
SOH CAH TOA
If you are trying to find the angles inside of a triangle, you can use the lengths to calculate the missing information:
Step 1: locate your angle of reference. This is the angle
that you are trying to solve. Label x.
Step 2: locate the right angle of the triangle. The side
(“leg”) opposite to the right angle is the
hypotenuse.
Step 3: locate the side opposite the hypotenuse. This
side (“leg”) is called the opposite.
Step 4: The remaining side is the adjacent.
© Community Learning Centre 2015 23
In order to find the angle of reference (“x”) you can use
the formula Sine. Once you have
Sine (beta) = opposite over hypotenuse
Sin ø = !""!!"
Step 5: Type in 2nd (function] sin and then the
opposite divided by the hypotenuse.
Step 6: Once you know two angles, you can calculate the third.
All triangles add up to: ______
Sine, in mathematics, is a “trigonometric function” of an angle. The sine of an angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to (divided by) the length of the longest side of the triangle (i.e. the
hypotenuse).
In mathematics, the beta function is a special function for trigonometry. Its symbol Β is a Greek capital β rather than the similar Latin capital B. It usually
means “unknown angle”
© Community Learning Centre 2015 24
Let’s go back to our tree example…
• We know that it is 530 feet from A to B • We know that angle a is 25 degrees. • Therefore we know that 90˚+ 25˚+ x˚ = 180˚ • The missing degree is 65˚
With this information, we now have what it takes to find the height of the tree! Step 1: decide which trig ratio you will use: tan, sin, cos
Step 2: TAN 25˚ = !""!#$%&!"#!$%&'(
Step 3: TAN 25˚ = !!"#
* you will need a calculator to find TAN 25. Make sure your calculator is in “degree mode.” To do this you type in 2nd Function 25 tan.
Step 4: 0.466 = !!"#
Step 5: Get x by itself. Multiple 0.466 by 530.
Step 6: x = 246.98 feet
Eurika! The height of the tree!
Well, not quite… remember to add your own height and any other missing height (distance from your feet to actual base of tree if the tree base is below you…)
© Community Learning Centre 2015 25
1. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find x. Solve for x.
2. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find∠C . Solve for∠C .
3. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find b. Solve for b.
© Community Learning Centre 2015 26
4. Which statement is incorrect? a. You can solve for the unknown side in any triangle, if you know
the lengths of the other two sides, by using the Pythagorean theorem.
b. The hypotenuse is the longest side in a right triangle.
c. The hypotenuse is always opposite the 90° angle in a right
triangle.
d. The Pythagorean theorem applies to all right triangles.
5. A roof is shaped like an isosceles triangle. The slope
of the roof makes an angle of 24 with the horizontal, and has an altitude of 3.5 m. Determine the width of the roof, to the nearest tenth of a meter.
6. The Leaning Tower of Pisa is held
up with wire. Without them the whole thing would fall over! What is the current angle of the Tower?