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MAC 1114

MAC 1114. Module 6 Trigonometric Identities II. Rev.S08. Learning Objectives. Upon completing this module, you should be able to: Apply the sum and difference identities for sine. Apply the sum and difference identities for tangent. Use the double-angle identities.

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MAC 1114

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  1. MAC 1114 • Module 6 • Trigonometric Identities II Rev.S08

  2. Learning Objectives • Upon completing this module, you should be able to: • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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

  4. Sine of a Sum or Difference What is the main difference between these two Identities? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  5. Tangent of a Sum of Difference What are the differences between these two Identities? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  6. Example of Applying the Sum Identities for Sine • Find an exact value for sin (105°). • sin 105 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  7. Example of Applying the Difference Identities for Sine • Find an exact value for sin 90° cos 135°− cos 90° sin 135° http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  8. Example of Writing Function as Expressions Involving Functions of θ • sin (30° + θ) • tan (45° + θ) Hint: Use the Sum or Difference Identities for Sine and Tangent. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  9. Example of Finding Function Values and the Quadrant of A + B • 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). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  10. Double-Angle Identities http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  11. Example of Using the Double-Angle Identities: Given cos(θ) = 3/5 and sin(θ) < 0, find sin(2θ). • Find the value of sinθ. • Use the double-angle identity for sine, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  12. Example of Using the Double-Angle Identities: Given cos(θ) = 3/5 and sin(θ) < 0, find cos(2θ). • Use any of the forms for cos to find cos 2θ. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  13. Example of Using the Double-Angle Identities: Given cos(θ) = 3/5 and sin(θ) < 0, find tan(2θ). • Find tan 2θ. or http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  14. Example of Deriving a Multiple-Angle Identity • Find an equivalent expression for cos(3x). • Solution http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  15. Product-to-Sum Identities http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  16. Example of Using a Product-to-Sum Identity • Write sin 2θcosθ as the sum or difference of two functions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  17. Sum-to-Product Identities http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  18. Example of Using Sum-to-Product Identity • Write cos 2θ−cos 4θas a product of two functions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  19. Half-Angle Identities http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  20. Example of Using the Half-Angle Identities • Find the sin (π/8) exactly. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  21. Example of Using the Half-Angle Identities • Find the exact value of tan 22.5° using the identity • Since 22.5° = replace A with 45°. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  22. Example of Using the Half-Angle Identities (Cont.) • Multiply the numerator and denominator by 2, than rationalize the denominator. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  23. Example of Simplifying Expression Using the Half-Angle Identities • Simplify: • Solution: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  24. What have we learned? • We have learned to: • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  25. 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 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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