Lesson 8: Using Trigonometry to Find Missing Sides

Module 1 Part 2: Trigonometry

 

Lesson 8: Using Trigonometry to Find Missing Sides

** 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**

 

The Leaning Tower of Pisa in Italy is a freestanding bell tower in the Italian city of Pisa and is known worldwide for its unintended tilt to one side. It is situated behind the Cathedral and is the third oldest structure in Pisa's Cathedral Square. The tower's tilt began during construction, caused by an inadequate foundation on ground too soft on one side to properly support the structure's weight. The tilt increased in the decades before the structure was completed, and gradually increased until the structure was stabilized (and the tilt partially corrected) by efforts in the late 20th and early 21st centuries.

 

When you have completed this lesson, you will be able to determine the lengths of the sides in the triangles that outline the tower.

 

Outcomes

 

At the end of this lesson, you will be able to find missing sides using sin, cos and tan in a right triangle.

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 use sine, cosine, and tangent ratios to find missing sides in right triangles?

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 located 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. Given the triangle below, complete the following:

a. the side opposite K is labelled as lower case _______
b. the side adjacent K is labelled as lower case _______
c. the side opposite L is labelled as lower case _______
d. the side adjacent L is labelled as lower case _______
e. the hypotenuse is labelled as lower case______

State the following as ratios using lower case letters:
f. sin K =
g. cos K =
h. tan K =
i. sin L =
j. cos L =
k. tan L =

2. State the sin, cos and tan for both angle X and Y in the triangle below. State your answers in both fractional and decimal form ( rounded to the nearest thousandth).

a. sin  X =
b. cos X =
c. tan X =
d. sin  Y =
e. cos Y =
f. tan  Y =

3. Use your calculator to find the following answers. Round your answers to the nearest thousandth.

a. sin 42^o =

b. cos 85^o =

c. tan 66^o =

4. Solve the following ratios for the missing variable.

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 Explore. It is also RECOMMENDED you work through the resources in the Refresher to review these important concepts before continuing through the lesson.

 

Refresher

Shape and Space: Trigonometry provides a good review before you begin solving for missing sides. On the right-hand side of the website, click on “Interactive.” When you are finished, choose “Video.”

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1: Part 2 Trigonometry

 

You have practiced finding the sine, cosine and tangent using triangles and by using your calculator. Let's use this knowledge to find missing sides in triangles.

Example

 

© City of Edmonton. Used under Creative Common - Attribution-Noncommercial-Share Alike 2.5 Canada Licence

 

Finding Missing Sides

Consider one of the triangles in the picture above:

How could we find the height of the monument?

We know that sin is defined as

In this example, we know that the angle is 50^o, and the hypotenuse is 62 m.
The side opposite the 50^o is unknown so we can call it x.

If we replace this information in the equation we get

Now we need to think about how to solve for the unknown value x......
We need to get rid of the 62 in the denominator on the right. So since it is currently divided, we 'undo' the division by multiplying each side by 62.

The 62's cancel on the right and we get

Now get a calculator out and finish
x = 62 x sin (50^o)
x = 47.5 m  (Note we round to a reasonable number of decimals)

Pick a different triangle from the picture.

How would we find the slant height ( hypotenuse) of the glass pyramid?

Start by identifying what sides we are given or asked to find.
85m is the adjacent side and ? is the hypotenuse. We will call the unknown h.

So we could use cos to solve for the unknown.

Substitute in the values we know:

To solve this one, we will need to do two steps:
Step 1: Multiply each side by h to get rid of it in the denominator.

Step 2: Divide both sides by cos 42^o, so that the h is on the left by itself.

Grab your calculator and evaluate.

h = 85÷cos(42) enter
h = 114.4 m ( rounded again)

 

Self-Check

 

Now that you have some experience with using trigonometry to find missing sides,  see how well you can answer the following questions.

 

SC 1. In the following triangle, find the missing side m. Round to the nearest hundredth.

SC 2. In the following triangle, find the missing side c. Round to the nearest hundredth.

SC 3. Find the length of t in the following diagram. Round to the nearest hundredth.

SC4: Given the right triangle, XYZ, with <Z =90^o, <X = 22^o, and z = 14 cm, find x. Round to the nearest hundredth.

 

Compare your answers.

Mathematics 10C Learn EveryWare © 2010 Alberta Education

                                            Module 1: Part 2 Trigonometry                                   

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Module 1 Part 2: Trigonometry

 

Lesson Assessment

 

Complete the lesson quiz posted under the Assess tab or by using the Quizzes link under the Activities 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 8 portion of the project.

 

Mathematics 10C Learn EveryWare © 2010 Alberta Education

Module 1 Part 2: Trigonometry

 

Lesson 8 Summary

 

In this lesson you examined the following key questions:

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

• How are the sine, cosine, and tangent ratios used to determine missing sides in right triangles?

In Lesson 9 you will be learning how to find missing angles in right triangles.

Mathematics 10C Learn EveryWare © 2010 Alberta Education