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### Section4.4post

Course: M 1330, Fall 2009
School: U. Houston
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Word Count: 439

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1330 4.4 1 4.4: M Trigonometric Expressions and Identities Algebraic Operations with Trig Functions We can manipulate expressions with trig functions using the same techniques we use when manipulating polynomials or rational functions. These techniques include distributing, collecting like terms, putting everything over a common denominator, and factoring. Perform the indicated operation Example 1: Simplify...

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1330 4.4 1 4.4: M Trigonometric Expressions and Identities Algebraic Operations with Trig Functions We can manipulate expressions with trig functions using the same techniques we use when manipulating polynomials or rational functions. These techniques include distributing, collecting like terms, putting everything over a common denominator, and factoring. Perform the indicated operation Example 1: Simplify [sec() + 5][sec() - 6] Example 2: Factor sin 2 - cos 2 Example 3: Factor sin 2 - 4 sin + 3 Example 4: Simplify csc - 3 sec Example 5: Simplify cos 2 + cos - 12 cos - 3 Example 6: Combine 3 5 + cos() sin( ) Identities: Recall: An identity is an equation that is true for all value(s) of the variable. Sometimes, you can use trig identities to help you simplify trig expressions. Here is a list of trig identities we all ready seen. tan( t ) = sin( t ) cos(t ) cot(t ) = cos(t ) sin( t ) M 1330 Reciprocal Identities: 4.4 2 csc(t ) = 1 , sin(t ) 0 sin( t ) sec(t ) = 1 , cos(t ) 0 cos(t ) cot(t ) = 1 , tan(t ) 0 tan(t ) Pythagorean Identities: sin 2 (t ) + cos 2 (t ) = 1 tan 2 (t ) + 1 = sec 2 (t ) 1 + cot 2 (t ) = csc 2 (t ) Opposite Angle Identities: sin(- t ) = - sin(t ) tan(- t ) = - tan(t ) sec(- t ) = sec(t ) cos(- t ) = cos(t ) csc(- t ) = - csc( t ) cot( - t ) = - cot( t ) Example7: Simplify using the opposite angle identities. a. sin( -30 ) Which of the following are equivalent? b. sin( - x ) - tan( x) Example 8: Simplify: tan( ) x csc( x ) Example 9: Simplify: csc 2 () - 1 1 - sin 2 () M 1330 4.4 sec () tan( ) + cot() 2 3 Example10: Simplify: Example 11: Simplify: (csc() - cot())(sec() + 1) You can use identities to help solve problems like the following: Example 12: If cot() = -5 , < < , find the exact values of cos and sin . 12 2 Proving identities: Your proof should look like this: (one side) = (......) = (........) = (.........) = (.........) = (other side) M 1330 Some helpful hints: 4.4 4 Start with the "ugliest" side. Draw a line under the equal sign so you don't move try to solve like an equation. Get common denominators. Convert everything to sine and cosine (this frequently works, but sometimes makes things worse). 5. Using sin 2 + cos 2 = 1 and other identities we'll learn later. 6. If you see 1 - sin , try multiplying by 1 + sin . If you see 1 + cos , t...

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