Module 5: Trigonometry Applications and Identities

 

In Try This 2 you determined a number of different identities. These are summarized in the following tables. The tables also include sum and difference identities for tangent and three different versions of the double-angle identity for cosine.

 

Sum Identities Difference Identities
sin (A + B) = sin A cos B + cos A sin B sin (A − B) = sin A cos B − cos A sin B
cos (A + B) = cos A cos B − sin A sin B cos (A − B) = cos A cos B + sin A sin B

 

Double-Angle Identities  
sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A
cos 2A = 2 cos2 A − 1
  cos 2A = 1 − 2 sin2 A

 

 



textbook

Read “Example 1” on page 301 of your textbook to see an example of how some of these identities can be used to simplify expressions.

 

 

Self-Check 1

 

Complete questions 1.a., 1.c., 1.e., 2.a., and 2.c. on page 306 of the textbook. Answer

So far you have seen that identities are statements that are always true when they are defined. Identities are useful in changing the form of an expression by substituting one side of the identity for the other. The following example shows how this can be done.



textbook

Read “Example 3” on pages 302 and 303 of the textbook. Pay attention when identity substitutions are used. For part a., notice that it is also possible to determine non-permissible values by solving sin 2x = 0 for x.

 

Self-Check 2
  1. Complete “Your Turn” from “Example 3” on page 303 of the textbook. Answer
  2. Complete questions 4.b., 4.d., 5.a., 5.d., and 12 on pages 306 to 307 of the textbook. Answer

It is possible to use the identities learned so far in this lesson to determine the exact value for some trigonometric expressions that use angles that are not multiples of 30°, 45°, or 60°. Complete Try This 3 to see how.

 

Try This 3

 

Consider the expression cos 75°.

  1. Write 75° as a sum or difference of angles that are multiples of 30°, 45°, or 60°.
  2. Use an identity to rewrite cos 75° in terms of the sum or difference you determined in question 1.
  3. Determine the exact value of your new expression. This is the exact value of cos 75°.
  4. Use a calculator to approximate cos = 75° and to approximate the exact value you determined in question 3. Do you get the same value?

course folder Save your responses in your course folder.

 

Share 2

 

With a partner or group, discuss the following question based on your solution to Try This 3.

 

Could you have rewritten 75° differently than you did in Try This 3 to determine the value of cos 75°? Explain.

 

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

cos 75° = cos (30° + 45°)