Rev.S08 MAC 1114 Module 6 Trigonometric Identities II

2 Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. Apply the sum and difference identities for sine. 2. Apply the sum and difference identities for tangent. 3. Use the double-angle identities. 4. Use the half-angle identities. 5. Evaluate expression with double-angle identities or half-angle identities. 6. Use product-to-sum identities. 7. Use sum-to-product identities. Click link to download other modules.

3 Rev.S08 Trigonometric Identities Click link to download other modules. - Sum and difference Identities for Sine and Tangent - Double-Angle Identities - Half-Angle Identities There are three major topics in this module:

4 Rev.S08 Sine of a Sum or Difference Click link to download other modules. What is the main difference between these two Identities?

5 Rev.S08 Tangent of a Sum of Difference Click link to download other modules. What are the differences between these two Identities?

6 Rev.S08 Example of Applying the Sum Identities for Sine Click link to download other modules. Find an exact value for sin (105 . sin 105

7 Rev.S08 Example of Applying the Difference Identities for Sine Click link to download other modules. Find an exact value for sin 90  cos 135  − cos 90  sin 135 

8 Rev.S08 Example of Writing Function as Expressions Involving Functions of  Click link to download other modules. sin (30  +  ) tan (45  +  ) Hint: Use the Sum or Difference Identities for Sine and Tangent.

9 Rev.S08 Example of Finding Function Values and the Quadrant of A + B Click link to download other modules. Suppose that A and B are angles in standard position, with sin A = 4/5,  /2 < A < , and cos B = −5/13,  < B < 3  /2. Find sin (A + B).

10 Rev.S08 Double-Angle Identities Click link to download other modules.

11 Rev.S08 Example of Using the Double-Angle Identities: Given cos(  = 3/5 and sin(  < 0, find sin(2 . Click link to download other modules. Find the value of sin . Use the double-angle identity for sine,

12 Rev.S08 Example of Using the Double-Angle Identities: Given cos(  = 3/5 and sin(  < 0, find cos(2 . Click link to download other modules. Use any of the forms for cos to find cos 2 .

13 Rev.S08 Example of Using the Double-Angle Identities: Given cos(  = 3/5 and sin(  < 0, find tan(2 . Click link to download other modules. Find tan 2 . or

14 Rev.S08 Example of Deriving a Multiple-Angle Identity Click link to download other modules. Find an equivalent expression for cos(3x). Solution

15 Rev.S08 Product-to-Sum Identities Click link to download other modules.

16 Rev.S08 Example of Using a Product-to-Sum Identity Click link to download other modules. Write sin 2  cos  as the sum or difference of two functions.

17 Rev.S08 Sum-to-Product Identities Click link to download other modules.

18 Rev.S08 Example of Using Sum-to-Product Identity Click link to download other modules. Write cos 2  − cos 4  as a product of two functions.

19 Rev.S08 Half-Angle Identities Click link to download other modules.

20 Rev.S08 Example of Using the Half-Angle Identities Click link to download other modules. Find the sin (  /8) exactly.

21 Rev.S08 Example of Using the Half-Angle Identities Click link to download other modules. Find the exact value of tan 22.5  using the identity Since 22.5  = replace A with 45 .

22 Rev.S08 Example of Using the Half-Angle Identities (Cont.) Click link to download other modules. Multiply the numerator and denominator by 2, than rationalize the denominator.

23 Rev.S08 Example of Simplifying Expression Using the Half-Angle Identities Click link to download other modules. Simplify: Solution:

24 Rev.S08 What have we learned? We have learned to: 1. Apply the sum and difference identities for sine. 2. Apply the sum and difference identities for tangent. 3. Use the double-angle identities. 4. Use the half-angle identities. 5. Evaluate expression with double-angle identities or half-angle identities. 6. Use product-to-sum identities. 7. Use sum-to-product identities. Click link to download other modules.

25 Rev.S08 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition Click link to download other modules.