Precalculus - Chapter 7 - Analytic Trigonometry

Precalculus Eleventh Edition Chapter 7 Analytic Trigonometry Copyright © 2020, 2016, 2012 Pearson Education, Inc....

Precalculus Eleventh Edition Chapter 7 Analytic Trigonometry Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 1 Section 7.4 Trigonometric Identities Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 2 Objectives 1. Use Algebra to Simplify Trigonometric Expressions 2. Establish Identities Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 3 Identically Equal, Identity, and Conditional Equation Definition Identically Equal, Identity, and Conditional Equation Two functions f and g are identically equal if f ( x)  g ( x) for every value of x for which both functions are defined. Such an equation is referred to as an identity. An equation that is not an identity is called a conditional equation. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 4 Identities (1 of 2) Quotient Identities sin  cos  tan   cot   cos  sin  Reciprocal Identities 1 1 1 csc   sec   cot   sin  cos  tan  Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 5 Identities (2 of 2) Pythagorean Identities sin 2   cos 2  1 tan 2   1 sec 2  cot 2   1 csc 2  Even-Odd Identities sin     sin  cos    cos  tan     tan  csc     csc  sec    sec  cot     cot  Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 6 Algebra Techniques Four basic algebraic techniques are used to establish identities: 1. Rewriting a trigonometric expression in terms of sine and cosine only 2. Multiplying the numerator and denominator of a ratio by a “well-chosen 1” 3. Writing sums of trigonometric ratios as a single ratio 4. Factoring Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 7 Example 1: Using Algebraic Techniques to Simplify Trigonometric Expressions (1 of 4) tan  a) Simplify by rewriting each trigonometric sec  function in terms of sine and cosine numbers. sin  1  cos  b) Show that  by multiplying the 1  cos  sin  numerator and denominator by 1  cos  cos 2   1 c) Simplify tan  cos   tan  by factoring. 1  sin u cot u  cos u d) Simplify  by rewriting the sin u cos u expression as a single ratio. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 8 Example 1: Using Algebraic Techniques to Simplify Trigonometric Expressions (2 of 4) sin  tan  sin  cos a) cos    sin  sec  1 cos 1 cos b) sin  sin  1  cos sin  (1  cos )    1  cos 1  cos 1  cos 1  cos 2  sin  (1  cos ) 1  cos   sin 2  sin  cos 2   1 (cos  1)(cos  1) cos  1 c)   tan  cos  tan  tan  (cos  1) tan  Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 9 Example 1: Using Algebraic Techniques to Simplify Trigonometric Expressions (3 of 4) 1  sin u cot u  cos u d)  sin u cos u 1  sin u cos u cot u  cos u sin u     sin u cos u cos u sin u cos u  sin u cos u  cot u sin u  cos u sin u  sin u cos u cos u  cot u sin u  sin u cos u Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 10 Example 1: Using Algebraic Techniques to Simplify Trigonometric Expressions (4 of 4) cos u  cos u d)  sin u cos u 2cos u  sin u cos u 2  sin u Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 11 Example 2: Establishing an Identity Establish the identity: csc  tan  sec  Start with the left side, because it contains the more complicated expression. Then use a reciprocal identity and a quotient identity. 1 sin  1 csc tan     sec sin  cos cos The right side has been reached, so the identity is established. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 12 Example 3: Establishing an Identity Establish the identity: sin 2      cos 2    1 Begin with the left side and, because the arguments are   , use Even–Odd Identities. 2 2 sin     cos     sin      cos    2 2 2 2 Even-Odd  sin    cos   Identities 2 2 sin    cos   1 Pythagorean Identity Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 13 Example 4: Establishing an Identity (1 of 2) sin 2     cos 2    Establish the identity: cos   sin  sin     cos    The left side contains the more complicated expression. Also, the left side contains expressions with the argument −θ, whereas the right side contains expressions with the argument θ. So start with the left side and use Even-Odd Identities. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 14 Example 4: Establishing an Identity (2 of 2) 2 2 sin     cos     sin      cos    2 2  sin     cos    sin     cos    2 2    sin    cos   Even-Odd Identities  sin   cos  2 2   sin    cos   Simplify.  sin   cos   sin   cos  sin   cos   Factor.  sin   cos   cos   sin  Simplify. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 15 Example 5: Establishing an Identity Establish the identity: 1  tan u tan u 1  cot u 1  tan u 1  tan u 1  tan u   1  cot u 1  1 tan u  1 tan u tan u tan u (1  tan u )  tan u  1 tan u Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 16 Example 6: Establishing an Identity (1 of 2) Establish the identity: sin  1  cos   2 csc  1  cos  sin  The left side is more complicated. Start with it and add. sin  1  cos sin 2   (1  cos ) 2   Add the 1  cos sin  (1  cos ) sin  quotients. sin 2   1  2cos  cos 2  Multiply out in  (1  cos ) sin  the numerator. (sin 2   cos 2  )  1  2cos  Regroup. (1  cos ) sin  Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 17 Example 6: Establishing an Identity (2 of 2) 2  2cos  sin 2   cos 2  1 (1  cos ) sin  2(1  cos ) Factor and cancel.  (1  cos ) sin  2  sin  2csc Reciprocal Identity Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 18 Example 7: Establishing an Identity Establish the identity: tan v  cot v 1 sec v csc v 1 Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 19 Example 8: Establishing an Identity (1 of 2) Establish the identity: 1  sin   cos cos 1  sin  Start with the left side and multiply the numerator and the denominator by 1  sin. (Alternatively, we could multiply the numerator and the denominator of the right side by 1 – sin.) Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 20 Example 8: Establishing an Identity (2 of 2) 1  sin  1  sin  1  sin  Multiply the numerator   cos cos 1  sin  and the denominator by 1  sin . 1  sin 2   cos (1  sin  ) cos 2  1 – sin 2 cos 2  cos (1  sin  ) cos  Cancel. 1  sin  Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 21 Guidelines for Establishing Identities It is almost always preferable to start with the side containing the more complicated expression. Rewrite sums or differences of quotients as a single quotient. Sometimes it helps to rewrite one side in terms of sine and cosine functions only. Always keep the goal in mind. As you manipulate one side of the expression, keep in mind the form of the expression on the other side. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 22