Lesson 7: Introduction to Trigonometry

Module 1 Part 2: Trigonometry

 

Lesson 7: Introduction to Trigonometry

 ** Always have your calculator in degree mode. Check by hitting the MODE button and looking at the third line. The word degree must be darkened**

Focus

 

treehouse: © Ewan Chesser/shutterstock; surveyor: © Joe Gough/shutterstock

 

As you have worked through the previous lessons, you have been adding shapes to create your own personal place. You have been asked to provide measurements and calculations for your shapes. Using a program like Google SketchUp makes it pretty easy to show measurements, since you can simply click on the tape measure and then drag the measure alongside your shapes. If only life were that simple!

 

Imagine building a tree house. There would be some measurements that are difficult to take, like the height of the tree—it’s not too easy (or safe) to climb a tree with a tape measure in your hands!

Luckily, there is a great mathematical concept called trigonometry. In the construction industry, surveying allows people to use trigonometry to calculate unknown lengths, so direct measurements are not required.

 

In this lesson you will explore trigonometry and answer questions such as, what does trigonometry mean and how does it work?

 

When you have completed this lesson, you will move on to Lesson 8 where you will learn how to solve for a missing side and then Lesson 9 where you will solve for a missing angle.

 

Outcomes

 

At the end of this lesson, you will be able to

• explain the meaning of sin, cos and tan in a right triangle.
• use the SIN, COS and TAN buttons on your calculator.

Lesson Questions

 

By the end of this lesson, you should feel comfortable solving the following questions:

• In what situations can the concepts of trigonometry be used to solve problems?

• How do I determine the sine, cosine, and tangent ratios for a given angle in a triangle?

• How do I use my calculator to find, sin, cos and tan for a given angle?

Lesson Completion and Assessment

 

As you work through each lesson, complete all the questions and learning activities in your binder using paper and pencil, clearly labeling your work (they refer to this as your course folder). These include the Are you Ready, Try This, Share and Self Check questions. Check your work if answers are provided. Remember that these questions provide you with the practice and feedback that you need to successfully complete this course.
Once you have completed all of the learning activities, take the Lesson Quiz. This is the assessment for each lesson and is locate under the Assess tab or by using the Quizzes link under the Activities block.

** Note – Share questions may have to be done on your own depending on your learning situation**

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Part 2: Trigonometry

 

 This section checks to see if you have the prerequisite knowledge and skills to be able to complete this lesson successfully.

Are You Ready?

Complete these questions in your course folder (binder). If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completing these exercises.

1. What is a right triangle?

2. In any triangle, what is the sum of the measures of the three angles?

3. Calculate the missing angle in the triangle below.

4. State the Pythagorean theorem, and explain when you would use it.

5. In the following triangle, we know the lengths of two sides. Use the Pythagorean theorem to calculate the length of the hypotenuse.

6. In the following triangle, we know the lengths of one side and the length of the hypotenuse. Use the Pythagorean theorem to calculate the length of the unknown side. Hint - re-arrange the formula first.

7. Find the length of the missing side or hypotenuse. Round answers to the nearest tenth.

a.

b.

Once you have completed these exercises to the best of your ability, use the provided answer link to check your work.

Answers

If you feel comfortable with the concepts covered in the questions, move forward to Discover. It is also RECOMMENDED you work through the resources in the Refresher to review these important concepts before continuing through the lesson.

 

Refresher

 

Are you able to tell if a triangle is a right triangle? Test your ability to do so by going to the multimedia piece titled “Right Triangle.” On the bottom of the website is an interactive definition of a right triangle.

 

Use the multimedia item titled “Pythagorean Theorem” to see another visual definition.

 

The multimedia piece titled “Exploring the Pythagorean Theorem” allows you to change the side lengths of a right triangle to see the effect on the length of the hypotenuse.

At the website, choose the “Interactive” button near the middle of the page. You will be presented with a visual explanation of the Pythagorean theorem.

Materials

• ruler

• protractor

• calculator

• graph paper

In addition, specific materials are required for the Going Beyond section where you have the opportunity to build a simple sundial. The materials you will require to build the sundial will depend on the design that you choose. You will find out what materials you need by doing an Internet search.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Part 2: Trigonometry

 

Discover

 

Watch and Listen

 

Have you ever wondered how pilots calculate a safe angle of descent when they are landing 747 planes? Go to “Exploring Trigonometry” and view the video, which talks about how trigonometry is used at airports. You will find the video on the right-hand side of the website.

 

Tip Labelling Triangles: In triangles, capital letters are used for angles and lower case letters are used for sides. The side opposite each angle is the same letter in lower case.

Try This 1 - 4

Go to  Try This 1 - 4 , print (or copy by hand) and complete each part on your page. Check your work using the link below.

Possible TT1-4 (Try This 1 - 4) Solutions

 

Share

 

Get together with another student or students (if possible) and compare your results.

• Compare the size of your triangle (the lengths of a, b, and c) with the other ones that have been posted on the discussion board. What do you notice?

• Compare the ratios , , and with others in your class. What do you notice?

• Can you make a generalization from your results?

Try This

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Part 2 Trigonometry

 

 

Glossary Terms

 

Retrieve your handout titled “Glossary Terms” from your binder. Add the following terms:

• adjacent side

• cosine ratio

• hypotenuse

• opposite side

• proportional

• Pythagorean theorem

• reference angle

• right triangle

• sine ratio

• solving a triangle

• tangent ratio

NAMING OF SIDES

The three sides of a right triangle are known by three different names. You need to be able to identify the opposite side, the adjacent side, and the hypotenuse.

 

The term hypotenuse may already be familiar to you. Have a look at the diagrams where the orange arrows each point to the hypotenuse. Can you see a pattern? If you were given a right triangle, how could you identify the hypotenuse?

 

 

The great thing about the hypotenuse is that it never changes locations—it is always directly across from the right angle.
Did you discover that pattern when you viewed the diagrams?

 

The two other sides that you need to identify are the opposite side and the adjacent side. The location of these other two sides will change, depending on which angle is being considered in the question.

 

 

 

 

For example, if you had the triangle to the right, you can see that the angle we need to find is labelled x. So, in this case, x is the reference angle. Can you tell which side is opposite from angle x?

 

 

 

If you said the bottom, you are correct!

 

See how the bottom side is opposite (or directly across) from angle x? In this case, we would label the bottom as the opposite side.

 

 

 

 

What about in the case to the left? Can you identify the opposite side if you are using angle y as the reference angle?

 

In this case, the left side is opposite from the reference angle.

 

Now that you can label the hypotenuse and the opposite side of a right triangle, all you need to do is label the remaining side as the adjacent side. You can also think of the adjacent side as the side between the reference angle and the 90° angle. Please review the examples below.

 

 

 

Learn and Listen

 

The interactive piece titled “Shape and Space” allows you to practise identifying the sides of a triangle by dragging and dropping the opposite, adjacent, and hypotenuse sides into their correct locations. You can find the interactive piece on the right-hand side of the website.

 

Self-Check

Complete the following self check using the link below to practice identifying the hypotenuse, the opposite side and the adjacent side.:

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Now that you know how to label the sides of a right triangle, we can practice more fully how to find sin, cos and tan using triangles and a calculator.

In the Discover section, you calculated three ratios which compare the lengths of the sides of a right triangle. Now that you know the names of the sides, you can apply them to the following triangle.

The ratios that you calculated in the Discover section were , and .

These ratios have names.

• The sine ratio is the ratio of the length of the side opposite the reference angle to the length of the hypotenuse.

• The cosine ratio is the ratio of the length of the side adjacent to the reference angle to the length of the hypotenuse.

• The tangent ratio is the ratio of the length of the side opposite the reference angle to the length of the side adjacent to the reference angle.

The definitions can be summarized by the following:

Notice that the ratios are based on the location of the reference angle.

A great way to remember the trigonometric ratios is by using the following:

SOH CAH TOA

If you chant SOH CAH TOA many times, the chant will stay in your head. (It sounds like “soak-a-toe-ah!”) Here’s what it means:

Module 1 Part 2: Trigonometry

 

 

Lesson Assessment

 

Complete the lesson quiz posted under the Assess tab or by using the Quizzes link under the Activitires block. Also, ensure your work in your binder (course folder) is complete. 

 

Project Connection **NOT ASSIGNED**

 

At this time, you should work on your Unit 1 Project. Go to the Unit 1 Project, and complete the Lesson 7 portion of the project.

 

Going Beyond

 

There are many uses of trigonometry. A sundial is a device that measures time based on the position of the Sun. A sundial is designed in such a way that the Sun casts a shadow from a sharp, straight edge onto a flat surface marked with lines that indicate the hours of the day.

 

In theory, a stick stuck in the ground could form the basis of a sundial. In reality, it’s not that simple. Earth’s axis is tilted, which means that the apparent movement of the Sun through the sky changes every day. If this isn’t accounted for, a sundial that tells perfect time today will be slightly wrong next week and very wrong next month.

 

See if you can find out how to build a sundial by doing a search on the Internet. Use the search terms “how to build a sundial.” Then look through a few of the websites to find a simple set of instructions.

 

(After you construct a real sundial, perhaps you will be interested in adding a virtual sundial into your unit project!)

 

In order to construct a sundial accurately, you will need to find the direction “due north” and you will also need to know the latitude of the town or city where you reside.

 

Note: Grande Prairie has a latitude of approximately 55°, Calgary is 51°, and Edmonton is 53°. You can go online to look up the latitude of your community. The multimedia piece Latitude Table shows a table that includes the latitude of many communities in Alberta.

 

Latitude & Longitude Table for Alberta

Place	Latitude Deg (N)	Latitude Min	Longitude Deg (W)	Longitude Min
Acadia Valley	51	12	110	04
Alix	52	27	113	13
Alsands	57	30	111	25
Barrhead	54	08	114	24

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Part 2: Trigonometry

 

Lesson 7 Summary

 

In this lesson you examined the following key questions:

• In what situations can the concepts of trigonometry be used to solve problems?

• How do I determine the sine, cosine, and tangent ratios for a given angle in a triangle?

• How do I use my calculator to find, sin, cos and tan for a given angle?

In Lesson 8, you will explore how to use trigonometry to find missing sides in right angled triangles.

Mathematics 10C Learn EveryWare © 2010 Alberta Education