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© Community Learning Centre 2015 1

Detective Cosine McTrig’s Private Journal of

Mathematical Enlightenment

Grade 10 Mathematics Lesson: Trigonometry

Community Learning Centre 2015


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


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


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˚


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


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?


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.


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


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:


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).

!! = !! = !! = !! + !!


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!


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


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


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.


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)


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?


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:


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


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


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:


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!


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.


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”


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…)


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.


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?

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Detective Cosine McTrig’s - WordPress.com Cosine McTrig’s ... SOHCAHTOA Pythagorean Theorem ... In order to measure the height of a tree you need