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# Note for MATH 1330 with Professor Flagg at UH

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15
PreCalSection4 4 Math 1330 Section 44 Trig 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 Example 1 Multiply sint9 3 sint9 6 5 cost9 sint9 Example 2 Combine Example 3 Simplify 9cost9 tant9 7cost9tant9 Example 4 Factor 16 cos2 9 81sin2 9 Exam le 5 Factor tan2 9 8tan 9 12 P cos2 9 cost9 12 Example 6 Factor cost9 3 PreCalSection4 4 2 Identities An identity is an equation with a variable that is true for all values of that variable The trig functions are de ned in such a way that there are many identities involving trig functions I will list the basic identities here and we will encounter more identities as we work with trig functions It is necessary to memorize the basic identities that I will list It is not necessary to memorize other identities but you need to be able to use them and prove they are true Basic Identities from the Definitions Note that all of these ratios are undefined when the function in the denominator is 0 I will not list the separate if X is not 0 conditions in this list to help keep the list easy to remember Definition Identities 1 tan 9 sm 6 cos 9 2 cot t9 cf8 6 sm 9 l 3 cot t9 tan 9 l 4 sec 9 cos 9 l 5 csct9 sm 9 Pythagorean Identities 6 sin2 t9cos2 9 1 7 tan2 61 sec2 9 8 lcot2 t9 csc2 9 Even 7 Odd Function Identities 9 sin t sint 10 cos t cost ll tan t tant 12 csc t csct l3 sec t sect l4 cot t cott The most important ones to memorize are l56910 The others can be derived directly from these identities PreCalSection4 4 Using Trig Identities to Evaluate Trig Functions Example 7 A cos 45 B tamg Example 8 If csct9 g and lt 9 lt 7239 nd the exact values for the other 5 trig functions for 9 Using Trig Identities to SimplifV Expressions Example 9 Simplify tanx cscx csc2 t9 l Exam le 10 Sim lif p p y l sin2t9 sec2 9 Exam le 11 Sim lif p p y tant9cott9 PreCalSection4 4 4 Example 12 Simplify csct9 cott9sect9 l Proving Identities You prove an identity by showing how you can algebraically manipulate one or both sides of the equation so that they both are the same Your proof should look like Left side Right Side This is NOT solving an equation it is using the other identities and definitions to simplify one side of the equation so that it looks exactly like the other side or simplify both sides so they look exactly alike Helpful Hints l 2 9939er 7 8 Start with the messiest side Draw a line under the equals sign or put a question mark over it so you remember not to move terms from one side to the other like solving an equation Get common denominators Convert everything to sine and cosine this often helps but not always Use the Pythagorean identities especially sin2 9 cos2 9 1 If you see l cos 9 in a fraction try multiplying both top and bottom by l cost9 and using the Pythagorean identity The same goes for 17 or 17 with other trig identities Try multiplying top an bottom by the other sign If you get stuck working on one side try the other Try substituting l for sin2 t9cos2 9 or substitute sin2 t9cos2 9 for 1 Example 13 Prove the following identity secx sinx tanx cosx PreCalSection4 4 Example 14 Prove the following identity cotAl tan2 A tam1 csc A Example 15 Prove the following identity sinx m cscx cotx Example 16 Prove the following identity csc4 x csc2 x cot4 x cot2 x Example 17 Prove the following identity 1 cosx cscx cotx sinx

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