Lesson 3

Lesson 3: Trigonometric Identities

Focus

Success in Formula 1 racing requires a powerful, lightweight, high-performance engine that is durable. For maximum speed and fuel efficiency, an aerodynamic design is used. The cockpit, also called the survival cell, is created out of carbon fibre and is designed with multiple safety features.

Suppose you own two race cars that are identical except for the colour. Would you expect the two cars to perform the same? Does it matter which car you race with? Is it reasonable to switch one car for the other and expect a similar result?

In this lesson you will learn about mathematical identities, which include expressions that look different but generally act the same.

Lesson Outcomes

At the end of this lesson you will be able to

• verify a trigonometric identity graphically and numerically
• determine non-permissible values for a trigonometric identity
• use the reciprocal, quotient, and Pythagorean identities to simplify an expression
• explain the difference between an identity and an equation

Lesson Questions

In this lesson you will investigate the following questions:

• How can you determine whether or not an expression is a trigonometric identity?
• How can trigonometric identities be used to simplify problems?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Before beginning this lesson you should be able to

• define the cosecant, secant, and cotangent ratios
• determine non-permissible values for a trigonometric function

Are You Ready?

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Rewrite each of the following using one of the three primary trigonometric ratios.
   1. sec θ Answer
   2. cot θ Answer
   3. csc θ Answer
2. Determine the values for which each of the following is undefined.
   1. tan x Answer
   2. csc x Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

Review the information on six common trigonometric ratios in Module 4, Lesson 4.

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

Try This 1

1. Use technology to plot the curve y = cos θ over the domain −2π ≤ θ ≤ 2π. Make a sketch of this function.
2. Use technology to plot the curve y = sin θ cot θ over the domain −2π ≤ θ ≤ 2π. Make a sketch of this function.
3. What do you notice about the two graphs?
4. Although the graphs appear similar, they are not exactly the same. Complete sin θ cot θ Table to determine the difference. Multiply sin θ by cot θ to determine sin θ cot θ in the table.

Save your answers in your course folder.

Share 1

With a partner or group, discuss the following questions based on the information in Try This 1.

1. When is the statement cos θ = sin θ cot θ true?
2. Predict all real values of θ for which the statement cos θ = sin θ cot θ is false.
3. When they are defined, are cos θ and sin θ cot θ ever unequal?

If required, save a record of your discussion in your course folder.

Explore

In Try This 1 you saw that y = cos θ and y = sin θ cot θ have the same values when both functions are defined. cos θ = sin θ cot θ is an example of a trigonometric identity. A trigonometric identity is a trigonometric equation that is true for all permissible values of the variable in the expressions on both sides of the equation.1

You have already experienced a number of trigonometric identities so far in this course, including reciprocal and quotient identities. Think of how they got their names.

It is often possible to predict whether a statement is an identity by verifying the statement numerically or graphically. Try This 2 explores this idea.

Try This 2

Consider the following equations:

1. Do you expect either of the equations to be a trigonometric identity? Explain.
2. Determine the non-permissible values for each equation.
3. Attempt to verify each potential identity by checking that both 40° and are solutions.
4. If you were to plot and y = cos x on the same graph and then plot and y = cos x on the same graph, which two graphs would you expect to overlap one another?
5. Plot the graphs in question 4 for domain −2π ≤ θ ≤ 2π. Was your prediction from question 4 correct?

Save your answers in your course folder.

Share 2

With a partner or group, discuss the following questions based on your answers from Try This 2.

1. Explain which of the two equations from Try This 2 you think is an identity.
2. Looking back at the definition for a trigonometric identity and using the information in Try This 2, is it possible to be sure the equation is an identity? Explain.

If required, save a record of your discussion in your course folder.

1 Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission. 

In Try This 2 you checked an identity by showing that particular values make the identity true and that the expressions on both sides of the equal sign appear to have the same graphs. This check is called a verification. A verification does not prove that an equation is an identity because it is not possible to verify a trigonometric equation for every possible value.

When simplifying, a possible strategy is to convert everything to sines and cosines and to divide out common factors where possible.

In Try This 3 you may have noticed that identities can be used to simplify expressions. A possible solution for Try This 3 is shown in Simplifying a Trigonometric Expression.

In Try This 4 you may have found that sin θ and cos θ are related by the identity sin2 θ + cos2 θ = 1 by drawing a diagram similar to the diagram shown. You also saw that 1 + tan2 θ = sec2 θ and 1 + cot2 θ = csc2 θ. As such, they are identities. These are called Pythagorean identities because each can be derived from the Pythagorean theorem, a2 + b2 = c2.

It is often useful to rearrange an identity to suit your needs. sin2 θ + cos2 θ = 1 can be rewritten as sin2 θ = 1 − cos2 θ or as cos2 θ = 1− sin2 θ. Pay attention to any restrictions on the identity if you rearrange it.

Watch “Pythagorean Identity Derivation” to see another derivation of these identities.

Source: Khan Academy (CC BY-NC-SA 3.0)

Connect

Lesson 3 Assignment

Lesson 3 Summary

In Focus you read about two race cars that were identical in every way other than colour. This analogy was used to describe trigonometric identities, equations with sides that look different but generally act the same. An identity is a mathematical equation that is true for any permissible value. In this lesson you worked with Pythagorean and reciprocal identities as well as some other less common identities. It is possible to verify an identity both graphically and numerically when given a potential identity.

Identities can be used to simplify some expressions; if one side of the identity appears in an expression, the other side of the identity can replace it. When simplifying using an identity, any restrictions on the identity also apply to the simplified expression.

In the next lesson you will learn some new trigonometric identities and use identities to help solve trigonometric equations.