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Rev.S08 MAC 1114 Module 6 Trigonometric Identities II

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

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3 Rev.S08 Trigonometric Identities http://faculty.valenciacc.edu/ashaw/ http://faculty.valenciacc.edu/ashaw/ 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:

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

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

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

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

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

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9 Rev.S08 Example of Finding Function Values and the Quadrant of A + B http://faculty.valenciacc.edu/ashaw/ 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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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