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  6 he number ofdaylight hours in Kajaani,Finland,varies from about 4.3 hr to 20.7 hr( Source  : The Astronomical Almanac  ).The functioncan be used to approximate the number ofdaylighthours H  on a certain day d  in Kajaani.We can use thisfunction to determine on which day ofthe year therewill be about 10.5 hr ofdaylight. This problem appears asExercise 54 in Exercise Set 6.5. H   d    7.8787 sin  0.0166 d   1.2723   12.1840 T  Trigonometric Identities,Inverse Functions,and Equations 6.1 Identities:Pythagorean and Sum and Difference 6.2 Identities:Cofunction,Double-Angle,and Half-Angle 6.3 Proving Trigonometric Identities 6.4 Inverses of the Trigonometric Functions 6.5 Solving Trigonometric Equations SUMMARY AND REVIEWTEST A P P L I C A T I O N 507 Copyright (c) 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley  508 Chapter 6ãTrigonometric Identities,Inverse Functions,and Equations State the Pythagorean identities.Simplify and manipulate expressions containing trigonometric expressions.Use the sum and difference identities to find function values. An identity  is an equation that is true for all  possible  replacements ofthevariables.The following is a list ofthe identities studied in Chapter 5. Basic Identities ,,,,,,,,,,In this section,we will develop some other important identities. Pythagorean Identities We now consider three other identities that are fundamental to a study of trigonometry.They are called the Pythagorean identities  .Recall that theequation ofa unit circle in the xy  -plane is.For any point on the unit circle,the coordinates x  and  y  satisfy this equa-tion.Suppose that a real number s  determines a point on the unit circlewith coordinates ,or .Then and .Substituting for x  and for  y  in the equation ofthe unit circlegives us the identity , Substituting cos s  for x  and sin s  for  y  which can be expressed as .sin 2   s   cos 2   s   1  cos s   2   sin s   2  1sin s  cos s  y   sin s x   cos s   cos s  ,sin s   x  ,  y   x  2   y  2  1cot x   cos x  sin x  tan x   sin x  cos x  cot x   1tan x  tan x   1cot x  tan   x     tan x  sec x   1cos x  cos x   1sec x  cos   x    cos x  sin     x     sin x  csc x   1sin x  sin x   1csc x  6.1 Identities:Pythagoreanand Sum andDifference s x  y  ( x  ,  y  ), or(cos s  , sin s  )(1, 0) x  2       y  2     1 Copyright (c) 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley  It is conventional in trigonometry to use the notation rather than.Note that .The identity gives a relationship between the sine and the cosine ofany real number s  .It is an important Pythagoreanidentity. We can divide by on both sides ofthe preceding identity:. Dividing by Simplifying gives us a second identity: . This equation is true for any replacement of   s  with a real number forwhich ,since we divided by .But the numbers for which(or ) are exactly the ones for which the cotangent and cosecant functions are not defined.Hence our new equation holdsfor all real numbers s  for which and are defined and is thus an identity.The third Pythagorean identity can be obtained by dividing by onboth sides ofthe first Pythagorean identity: Dividing by . Simplifying  The identities we have developed hold no matter what symbols areused for the variables.For example,we could write ,,or . Pythagorean Identities ,,1  tan 2   x   sec 2   x  1  cot 2   x   csc 2   x  sin 2   x   cos 2   x   1sin 2   x   cos 2   x   1sin 2      cos 2      1sin 2   s   cos 2   s   1  tan 2   s   1  sec 2   s  cos 2   s  sin 2   s  cos 2   s   cos 2   s  cos 2   s   1cos 2   s  cos 2   s  csc s  cot s  sin s   0sin 2   s   0sin 2   s  sin 2   s   0 1  cot 2   s   csc 2   s  sin 2   s  sin 2   s  sin 2   s   cos 2   s  sin 2   s    1sin 2   s  sin 2   s  sin 2   s   cos 2   s   1     2  y   112  y      sin x  2     sin ( x    ⋅   x  )      x  2 sin 2   s   sin s  2  sin s   2 sin 2   s  Section 6.1ãIdentities:Pythagorean and Sum and Difference 509    2    2  x  y   112  y      (sin x  ) 2     (sin x  )(sin x  )    Copyright (c) 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley  It is often helpful to express the Pythagorean identities in equivalentforms. Simplifying Trigonometric Expressions We can factor,simplify,and manipulate trigonometric expressions in thesame way that we manipulate strictly algebraic expressions. EXAMPLE 1 Multiply and simplify:. Solution Multiplying Recalling the identities and and substituting Simplifying  There is no general procedure for manipulating trigonometric ex-pressions,but it is often helpful to write everything in terms ofsines andcosines,as we did in Example 1.We also look for the Pythagorean iden-tity,,within a trigonometric expression. EXAMPLE 2 Factor and simplify:. Solution Removing a common factorUsing  cos 2   x  sin 2   x   cos 2   x   1  cos 2   x    1   cos 2   x     sin 2   x   cos 2   x   sin 2   x   cos 2   x   cos 4   x  sin 2   x   cos 2   x   cos 4   x  sin 2   x   cos 2   x   1  sin x   1 sec x   1cos x  tan x   sin x  cos x   cos x   sin x  cos x   cos x   1cos x   cos x   tan x   cos x   sec x  cos x     tan x   sec x   cos x     tan x   sec x   510 Chapter 6ãTrigonometric Identities,Inverse Functions,and Equations P YTHAGOREAN I DENTITIES E QUIVALENT F ORMS tan 2   x   sec 2   x   11  sec 2   x   tan 2   x  1  tan 2   x   sec 2   x  cot 2   x   csc 2   x   11  csc 2   x   cot 2   x  1  cot 2   x   csc 2   x  cos 2   x   1  sin 2   x  sin 2   x   1  cos 2   x  sin 2   x   cos 2   x   1 Study Tip The examples in each sectionwere chosen to prepare you forsuccess with the exercise set.Study the step-by-step annotated solutions of theexamples,noting thatsubstitutions are highlighted in red.The time you spendunderstanding the examples willsave you valuable time when  you do your assignment. Copyright (c) 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
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