book 9: triangles - gssd blogsblogs.gssd.ca/csmith/files/2015/06/book-9-student-triangles.pdf ·...

Math 21 Home

Book 9: Triangles

Name: ______________________________

Start Date: ______________ Completion Date: ________________

Book 9: Math 21 Home- Triangles Edited April 2015

Year Overview:

Earning and Spending Money

Home Travel and

Transportation Recreation and Wellness

1. Budget 2. Personal Banking 3. Interest 4. Consumer Credit 5. Major Purchases

6. Scale Drawings & Ratios

7. Area & Volume 8. Angles 9. Triangles 10. Slope & Elevation

11. Travel Project 12. Puzzles and Games 13. Understanding

Statistics 14. Budgeting Recreation

Topic Overview There is a lot of mathematics that can help you understand, design, and create things at home. Scale drawings help you to design decks and buildings, or read architectural drawings. Ratios not only help you to interpret scale drawings, you also see them in TVs and computer monitors. This section of the home unit is designed to help you understand and create scale drawing and understand the ratios around you. Suggested Timeframe: 8 Hours

Outcomes

Overlapping Outcomes in Scale Drawings and Ratios

M21.1 Extend and apply understanding of the preservation of equality by solving problems that involve the manipulation and application of formulae within home, money, recreation, and travel themes.

Theme Specific Outcomes

M21.6 Demonstrate understanding of primary trigonometric ratios (sine, cosine, and tangent).

2


Contents Topic Overview.................................................................................................................. 2

Outcomes ....................................................................................................................... 2

Overlapping Outcomes in Scale Drawings and Ratios ....................................... 2

Theme Specific Outcomes ....................................................................................... 2

Glossary of Terms ........................................................................................................... 4

9.1 Right Triangles .............................................................................................................. 4

Sides in a Right Triangle ................................................................................................ 5

9.1 Practice Your Skills: Labelling Right Angle Triangles ........................................... 6

9.2 Ratios in Trigonometry ................................................................................................ 7

Discuss the Ideas: Investigating Trigonometry Ratios ............................................... 8

A. The Three Primary Trig Ratios ................................................................................... 9

B. Using Formula Triangles .......................................................................................... 10

9.3 How To Use Ratios to Solve Triangles ...................................................................... 11

9.3 Practice Your Skills – Calculating Triangle Ratios .............................................. 13

9.3 A. The Tangent Ratio ............................................................................................ 14

9.3A. Practice Your Skills – The Tan Ratio .............................................................. 14

9.3B. The Sine and Cosine Ratios .............................................................................. 18

9.3B. Practice Your Skills – Sine and Cosine Ratios .............................................. 19

9.3 Practice Your Skills: Solving Triangles .............................................................. 21

9.3 Practice Your Skills – Solving Triangles............................................................. 22

Student Evaluation .......................................................................................................... 23

Learning Log .................................................................................................................... 25

3


Glossary of Terms

9.1 Right Triangles One angle of a right triangle must = 90⁰.

In the diagram ∠C is the right angle, noted by the square in the corner.

Note that angles are labeled with UPPER CASE letters and the sides, or legs, are labeled in lower case letters to correspond with the angles that are across from them.

Label the 3 sides of the following triangles by their lower case letter names. Then solve the missing angle.

A

C B side a

side

b

90⁰

T

V

U ______

______

______

Q R

P

______ ______

______

47⁰ A

C B

_____34⁰

G

I H

______

13⁰

______

4


A

C B Side a is Adjacent

Side

b is

O

ppos

ite

Sides in a Right Triangle The sides of a right triangle are labelled according to where the right angle is. The first step is to identify the right angle, and then label the sides according to the following:

• Hypotenuse: In a right triangle, the longest side is called the hypotenuse and is always opposite or across from the right angle.

• Opposite: In a right triangle, the opposite side is across from the given angle or angle of interest.

• Adjacent: In a right triangle, the adjacent side is beside the given angle or angle of interest.

In the diagram below ∠ B is the given angle or the angle of interest and the sides are labeled in relation to ∠ B.

Side a is adjacent to ∠ B

Side b is opposite to ∠ B

5


9.1 Practice Your Skills: Labelling Right Angle Triangles From the given angle, label the 3 sides of the following triangles as opposite (opp), adjacent (adj), and hypotenuse (hyp):

T

V

U ______

______

______

Q R

P

______ ______

______

47⁰

A

C B

_____

_____34⁰

G

I H

______

13⁰

6


9.2 Ratios in Trigonometry A Ratio is the relationship between any two sides of a triangle and can be useful for a variety of calculations.

For example, compare the 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝐻𝐻𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝑂𝑂𝐻𝐻𝑂𝑂𝑂𝑂

ratio of the two triangles below:

Ratio of 𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝐻𝐻𝑂𝑂

= Ratio of 𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝐻𝐻𝑂𝑂

=

• Notice how the ratio changes when the size of the angle changes. • Notice also how the steepness of side c changes in the drawing.

The ratios of the sides of a triangle are very useful when we want to know the lengths of the sides of a triangle without measuring them all.

A

C B

10 cm

7 cm

60⁰

A

C B

10

3 cm

23⁰

7


Discuss the Ideas: Investigating Trigonometry Ratios 1. Choose any angle between 10° and 90°, except not 44, 45, or 46. 2. On a separate piece of paper, draw 5 different sized right angle triangles

that contain your chosen angle. (Be sure to draw these fairly large so that they are easier to work with.) For example, the following triangles all contain the angle 42°:

3. Label your right angle and chosen angle on all of the triangles you create and the sides of each triangle as hypotenuse, opposite, or adjacent.

4. Measure each side of every triangle, and fill in the table below.

5. Using your calculator, calculate the ratios of the triangle sides in the columns shown.

Tria

ngle

Opp

osite

(O)

Ad

jace

nt (A

)

Hypo

tenu

se

(H) 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂

𝐻𝐻𝐻𝐻𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝑂𝑂𝐻𝐻𝑂𝑂𝑂𝑂

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑂𝑂𝐻𝐻𝑂𝑂

𝐻𝐻𝐻𝐻𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝑂𝑂𝐻𝐻𝑂𝑂𝑂𝑂

𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑂𝑂𝐻𝐻𝑂𝑂

1 2 3 4 5

Average Ratios

6. Calculate the average for each ratio. 7. Use your calculator to find the following ratios for your chosen angle:

Chosen Angle: Sine Cosine Tangent

• To find the Sine of an angle, use the Sin button on your calculator. For Cosine, use Cos, and for Tangent use Tan.

8. What do you notice about your calculator values and the averages of your ratios?

8


𝑆𝑆𝑂𝑂𝐻𝐻 𝐴𝐴 = 𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝐻𝐻𝑂𝑂

𝐶𝐶𝑂𝑂𝑂𝑂 𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐻𝐻𝐻𝐻𝑂𝑂

𝑇𝑇𝐴𝐴𝐻𝐻 𝐴𝐴 = 𝑂𝑂𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴𝐴𝐴

A. The Three Primary Trig Ratios As you discovered, the sides of a right triangle with a given angle give you a ratio that does not change, regardless of the size of the triangle.

There are 3 specific ratios that are useful in any right triangle.

The Sine Ratio

The Sine ratio = 𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂

The Cosine Ratio

The Cosine ratio = 𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒐𝒐𝒉𝒉𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒉𝒉𝒉𝒉𝒐𝒐𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂

The Tangent Ratio

The Tangent ratio = 𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂𝑙𝑙𝑂𝑂𝐻𝐻𝑙𝑙𝑂𝑂ℎ 𝑂𝑂𝑜𝑜 𝑂𝑂ℎ𝑂𝑂 𝑂𝑂𝑂𝑂𝑠𝑠𝑂𝑂 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒐𝒐𝒉𝒉𝒐𝒐 𝑂𝑂𝑂𝑂 𝑂𝑂ℎ𝑂𝑂 𝑙𝑙𝑂𝑂𝑔𝑔𝑂𝑂𝐻𝐻 𝑎𝑎𝐻𝐻𝑙𝑙𝑙𝑙𝑂𝑂

Opp

osite

Adjacent

Opp

osite

Adjacent

9


𝐻𝐻𝐻𝐻𝑂𝑂 = 𝑂𝑂𝑂𝑂𝑂𝑂𝑆𝑆𝑂𝑂𝐻𝐻 𝐴𝐴

B. Using Formula Triangles Solving equations with 3 parts can be easy if you manipulate the formula before you put your known values in. When you have a 3 part equation like the Sine, Cosine, and Tangent ratios, you can create something called a formula triangle to help you. Here is an example of how to do this.

The value on the top of the ratio fills in the top of the formula triangle, and the other two entries fill in the bottom two spaces in the formula triangle.

To use a formula triangle, you simply ask yourself “What is this question asking me to calculate?” This becomes the value that becomes the left hand side of your new equation. Cover that entry up in the formula triangle, and you can see what is left to create the right hand side of your new equation.

For example, if you are wanting to calculate the Hyp of a triangle, and know the angle and the Opp side, then:

Once you have the formula changed so that the unknown is on the left side, it is easier now to substitute known values in and calculate.

SinA Hyp

Opp

10


9.3 How To Use Ratios to Solve Triangles You can use these ratios to help you find the measure of sides or angles in a

What is the length of side X?

Before calculating, we need to identify and label the sides of the triangle relative to the angle that is given.

We can see that we know the Adjacent side, and want to know the Opposite side. We neither know nor need to know the Hypotenuse. Looking at our three trigonometric ratios:

𝑆𝑆𝑂𝑂𝐻𝐻 𝐴𝐴 = 𝑂𝑂𝑂𝑂𝑂𝑂𝐻𝐻𝐻𝐻𝑂𝑂

𝐶𝐶𝑂𝑂𝑂𝑂 𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐻𝐻𝐻𝐻𝑂𝑂

𝑇𝑇𝐴𝐴𝐻𝐻 𝐴𝐴 = 𝑂𝑂𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴𝐴𝐴

We can see that the ratio that involves Opposite and Adjacent is Tan. Using the Tan Ratio. We can think of this in a formula triangle:

𝑂𝑂𝑂𝑂𝑂𝑂 = (𝑇𝑇𝐴𝐴𝐻𝐻 𝐴𝐴)(𝐴𝐴𝐴𝐴𝐴𝐴) 𝑂𝑂𝑂𝑂𝑂𝑂 = (𝑇𝑇𝐴𝐴𝐻𝐻 35°)(42𝑚𝑚) 𝑂𝑂𝑂𝑂𝑂𝑂 = (0.7002)(42𝑚𝑚)

𝑂𝑂𝑂𝑂𝑂𝑂 = 29.4𝑚𝑚 The length of the unknown side is 29 4 m

Note: Make sure your

calculator is in DEGREE

mode.

11


The tan-1 button is usually found when you SHIFT tan.

triangle.

There is a button on your calculator that looks like TAN-1. This is called the Inverse TAN. You can use it to find out the size of Angle A, or the Angle of Elevation.

If the slope = 3.5, then what is the Angle of Elevation?

Slope = Tan A

3.5 = Tan A

Tan-1 3.5 = A

75.0° = A

Therefore, the angle of elevation for a slope of 3.5 is 75°.

12


9.3 Practice Your Skills – Calculating Triangle Ratios

13


Tan A = opp adj

9.3 A. The Tangent Ratio The short form of the formula looks like this:

1. In the ∆ ABC, identify using lower case letters a. The side opposite ∠A ______ b. The side adjacent to ∠A ______ c. Tan A (opp

adj) ______

d. The side opposite ∠ B ______ e. The side adjacent to ∠ B ______ f. Tan B ______

2. In the ∆ DEF, identify using lower case letters

a. The side opposite ∠D ______ b. The side opposite ∠ E ______ c. The side adjacent to ∠D ______ d. The side adjacent to ∠ E ______ e. Tan D ______ f. Tan E ______

3. Use the tangent function (tan) on your calculator to find the value of each of

the following to 4 decimal places. tan 11⁰ ________ tan 71⁰ ________ tan 24⁰ ________ tan 87⁰ ________

4. Use the inverse tangent function (tan-1) to find the size of angle A to two

decimal places for the given tangent ratio. tan A = 0.3164 ________ tan A = 1.6250 ________

tan A = 0.5000 ________ tan A = 1.8642 ________

9.3A. Practice Your Skills – The Tan Ratio Solve for x in each of the following triangles. Give each answer to two decimal places.

Tan A = opp adj

A

C B

c

a

b

D

F E

14


Show your work here: 1.

2.

3.

4.

22° x

43°

38

x

37°

19

x

14

15

x

2

15


5. The wheelchair ramp is built to rise 1 meter for every 12 meters. Find the angle of the ramp.

6. The peak of the roof of a building is 4 meters above the height of the walls. If the width of the walls is10 meters, find the size of the angle between the roof and the walls.

12

1x

10 m

x 4 m

16


7. A ladder leaning against the house makes an angle of 65⁰ with the ground. If the bottom of the ladder is 1.2 m from the house, how high on the wall is the ladder? Draw a picture and then solve.

8. You need to make steps to go from your lawn to your deck. You want to start your stairs 120 cm along the lawn from the edge of the deck and the deck is 90 cm above the ground. Make a drawing and then calculate the angle of your stairs.

17


9.3B. The Sine and Cosine Ratios Formulas:

1. In the ∆ PQR, identify using lower case letters a. The side opposite ∠ P ______ b. The side opposite ∠Q ______ c. The hypotenuse side ______ d. Sin P ______ e. Sin Q ______

2. In the ∆ MNO, identify using lower case letters

a. The side adjacent ∠M ______ b. The side adjacent ∠N ______ c. The hypotenuse side ______ d. Cos M ______ e. Cos N ______

3. Use the sine (sin) and cosine (cos) function on your calculator to find the

value of each of the following to 4 decimal places. sin 13⁰ ________ cos 57⁰ ________ sin 62⁰ ________ cos 49⁰ ________

4. Use the inverse sine or cosine function (sin-1/cos-1) to find the size of angle A to two decimal places for the given ratio.

sin A = 0.6427 ________ cos A = 0.8521 ________

sin A = 0.5000 ________ cos A = 0.6428 ________

sin A = opp hyp

cos A = adj ℎ𝐻𝐻𝑂𝑂

P

R Q

O

M

18


9.3B. Practice Your Skills – Sine and Cosine Ratios Solve for x in each of the following triangles. Answer to 2 decimal places.

1.

2.

3.

4.

5.

sin A = opp hyp

cos A = adj hyp

SinA Hyp

Opp

Cos Hyp

Adj

11 m

x 30⁰

34 m

x

38⁰

75 m

x 58⁰

9

x

26° 14

x

19


6. A roadway rises at an angle of 6° to the horizontal. If you drive 20 kilometers up the hill, by how much has your altitude increased?

7. To make sure you do not get water in your basement, you hire landscapers to grade your lawn. Before they started and the ground was level, the distance from your house to the street was 12 m. After they are done, you notice that the ground is higher up against your house and it is now 13 m down a slight hill to the street. Draw a picture and calculate the angle of elevation.

8. You are putting together a play set for your back yard. The set comes with a 2.5 m ladder and two options for the slide. One slide is 4 m long and the other is 6 m long. Draw both slides and calculate the angle of elevation for each. Which one is steeper?

x 6⁰

20


9.3 Practice Your Skills: Solving Triangles

21


9.3 Practice Your Skills – Solving Triangles

22


Student Evaluation

Insufficient Evidence (IE)

Developing (D) Growing (G) Proficient (P) Exceptional (E)

Student has not demonstrated the criteria below.

Student has rarely demonstrated the criteria below.

Student has inconsistently demonstrated the criteria below.

Student has consistently demonstrated the criteria below.

Student has consistently demonstrated the criteria below. In addition they have shown their understanding in novel situations or at a higher level of thinking than what is expected by the criteria.

Proficient Level Criteria IE D G P E

M21.1 Extend and apply understanding of the preservation of equality by solving problems that involve the manipulation and application of formulae within home, money, recreation, and travel themes.

a. I can prove whether given forms of the same formula are equivalent and justify the conclusion.

b. I can describe, using examples, how a given formula is used in a home.

c. I can create, solve, and verify the reasonableness of solutions to questions that involve a formula.

e. I can solve questions that involve the application of a formula that:

• Does not require manipulation • Does require manipulation

23


Proficient Level Criteria IE D G P E

M21.6 [WA 10.8 and WA 20.9] Demonstrate understanding of primary trigonometric ratios (sine, cosine, and tangent)

a. I can describe the properties of a triangle

b. I can determine the missing angle in a triangle.

c. I can observe a set of similar right triangles and consider and draw conclusions about the ratios of the lengths, with respect to one acute angle of the:

• side opposite to the side adjacent • side opposite to the hypotenuse • side adjacent to the hypotenuse

d. I can use formula for the primary trigonometric ratios (cosine, tangent, and sine).

e. I can describe, using examples, how I would use a trigonometric formula in the home context.

f. I can figure out solutions to questions that involve primary trigonometric ratios to determine if they are reasonable and explain the reasoning.

24


Learning Log Date Starting Point Ending Point

25

book 9: triangles - gssd blogsblogs.gssd.ca/csmith/files/2015/06/book-9-student-triangles.pdf ·...

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