MAC 1114 Module 6 Trigonometric Identities II Rev

MAC 1114 Module 6 Trigonometric Identities II Rev. S 08

Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. 7. Apply the sum and difference identities for sine. Apply the sum and difference identities for tangent. Use the double-angle identities. Use the half-angle identities. Evaluate expression with double-angle identities or half-angle identities. Use product-to-sum identities. Use sum-to-product identities. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 2

Trigonometric Identities There are three major topics in this module: - Sum and difference Identities for Sine and Tangent - Double-Angle Identities - Half-Angle Identities Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 3

Sine of a Sum or Difference n What is the main difference between these two Identities? Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 4

Tangent of a Sum of Difference What are the differences between these two Identities? Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 5

Example of Applying the Sum Identities for Sine n n Find an exact value for sin (105°). sin 105 Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 6

Example of Applying the Difference Identities for Sine n Find an exact value for sin 90° cos 135° − cos 90° sin 135° Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 7

Example of Writing Function as Expressions Involving Functions of θ n sin (30° + θ ) n tan (45° + θ ) Hint: Use the Sum or Difference Identities for Sine and Tangent. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 8

Example of Finding Function Values and the Quadrant of A + B n Suppose that A and B are angles in standard position, with sin A = 4/5, π/2 < A < π, and cos B = − 5/13, π < B < 3π/2. Find sin (A + B). n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 9

Double-Angle Identities n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 10

Example of Using the Double-Angle Identities: Given cos(θ ) = 3/5 and sin(θ ) < 0, find sin(2θ ). n Find the value of sinθ. Rev. S 08 n Use the double-angle identity for sine, http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 11

Example of Using the Double-Angle Identities: Given cos(θ ) = 3/5 and sin(θ ) < 0, find cos(2θ ). n Use any of the forms for cos to find cos 2θ. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 12

Example of Using the Double-Angle Identities: Given cos(θ ) = 3/5 and sin(θ ) < 0, find tan(2θ ). n Find tan 2θ. or Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 13

Example of Deriving a Multiple-Angle Identity n n Find an equivalent expression for cos(3 x). Solution Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 14

Product-to-Sum Identities Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 15

Example of Using a Product-to-Sum Identity n Write sin 2θ cosθ as the sum or difference of two functions. n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 16

Sum-to-Product Identities Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 17

Example of Using Sum-to-Product Identity n Write cos 2θ − cos 4θ as a product of two functions. n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 18

Half-Angle Identities n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 19

Example of Using the Half-Angle Identities n Find the sin (π/8) exactly. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 20

Example of Using the Half-Angle Identities n Find the exact value of tan 22. 5° using the identity n Since 22. 5° = Rev. S 08 replace A with 45°. http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 21

Example of Using the Half-Angle Identities (Cont. ) n Multiply the numerator and denominator by 2, than rationalize the denominator. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 22

Example of Simplifying Expression Using the Half-Angle Identities n Simplify: Rev. S 08 n Solution: http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 23

What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. 7. Apply the sum and difference identities for sine. Apply the sum and difference identities for tangent. Use the double-angle identities. Use the half-angle identities. Evaluate expression with double-angle identities or half-angle identities. Use product-to-sum identities. Use sum-to-product identities. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 24

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, 8 th Edition Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 25