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by: Danielle Moore


Danielle Moore
GPA 3.61


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This 4 page Class Notes was uploaded by Danielle Moore on Saturday September 12, 2015. The Class Notes belongs to Uni Stu 6 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/201930/uni-stu-6-university-of-california-irvine in University Studies at University of California - Irvine.




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Date Created: 09/12/15
Chapter 6 Trigonometric Identities and Conditional Equations 61 Basic Identities and Their Use 62 Sum Difference and Confunction Identities 63 DoubleAngle and HalfAngle Identities 64 ProduceSum and SumProduct Identities 65 Trigonometric Equations Identity An equation in one or more variables is said to be an identity if the left side is equal to the right side for all replacements of the variables for which both sides are de ned For example7 the equation 272m78x74x2 is an identity Conditional equation An equation in one or more variables is said to be a conditional equation if the equation holds only for certain values of z and not for all values for which both sides are de ned For example7 the equation m2 7 2 m 7 8 0 is a conditional equation 61 Basic Identities and Their Use Basic Identities Basic Trigonometric Identities For x any real number in all restricted so that both sides of an equation are de ned Reciprocal Identities 1 cscm i secm cotz7 s1n z cos x tan z Quotient Identities sin z cos x cos x sin z Identities for Negatives sin7z isinz cos7m cosz tan7z itanz Pythagorean Identities sin2zcos2z1 tan2m1sec2z 1cot2mcsc2z i 2 i2 tan21 smz 1 s1n z1 cos x cos2 x sin2 z cos2 x sin2 z cos2 x cos2 x cos2 cos2 x 1 72 secZ ac cos no 2 cos x 2 sin2 z cos2 x 1 cot z 1 i i 2 i 2 sin no sin no sin no sin2 z cos2 x sin2 z cos2 x sin2 z sin2 sin2 z f csc2 no sin ac Establishing Other Identities Identities are established in order to convert one form to an equivalent form that may be more useful To verify an identity means to prove that both sides of an equation are equal for all replacements of the variables for which both sides are de ned Such a proof might use basic identities7 factoring7 combining and reducing fractions7 and so on The methods to verify certain identities are not unique To become pro cient in the use 3 of identities it is important that you work out many problems on your own Example Verify the identities A cosztanz sin z B sec7m sec x C cotzcosz sinz cscm Solution To verify an identity we proceed by starting with the more com plicated of the two sides and transform that side into the other side in one or more steps using basic identities algebra or other established identities sin z cos mtan z cos x COS sec7m cos as cotmcoszs1nz coszs1nz sin 95 cos2 as s1n z sin 95 cos2 x sin2 z sin z sin z csc as D Suggested steps in verifying identitiesPage 454 of the textbook 1 Start With the more complicated side of the identity and transform it into the simpler side 2 Try algebraic operations such as multiplying factoring combining fractions and splitting fractions 3 If other steps fail express each function in terms of sine and cosine functions and then perform appropriate algebraic operations 4 At each step keep the other side of the identity in mind This often reveals what you should do in order to get there Review of the algebraic identities l ab2 a2 2abb2 2 ab2a22abb2 3 a2b2abab 4 a3b3aba2abb3 5 a3b3aba2abb3 a 0 6 WVT a c ac 7777 12 b b a c adcb lt8gt77 b d7 dbd b a c a 0 9 gig ac bd Warm Example verify the identities 1sinm cosz A 7A 2 seem cosz 1s1nm sin2z2 sinz1 1sinm B cosZz 1sinm tanzcotm C 12 coszz tanm cot z Solution 1 sinz cosz 7 1 sinz2 coszz cosz 1sinm 7 cosz1sinm 12 sinzsin2mcos2z cosz1sin 12 sinz1 cosz 1 sinz 22 sinz cosz 1 sinz 21sinz cosz 1 sinz 2 cosm 2 sec z


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