Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
A2HCh0705 Multiple Angles and Product to Sum Formulas Homework and Reading Read p569– 576 (1) HW p577 #1– 53 odd (2) HW p578 #55 – 81 odd #95–109 odd, 121 Goal Multiple-Angles Formulas Students use multiple-angle formulas, power-reducing formulas, half-angle formulas, productto-sum formulas, and sum-to-product formulas to rewrite and evaluate trigonometric functions, Graph y = cos 4 x ! sin 4 x on [ 0, 2" ) Example Multiple – Angles Formula Double – Angle Formulas cos 2u = cos 2 u ! sin 2 u = 2 cos 2 u ! 1 2 tan u tan 2u = 1 ! tan 2 u Example Simplify cos x ! sin x 4 sin 2u = 2 sin u cosu ( 4 2 = cos 2x 1 2 " 5" x= , 6 6 solve -! 2 tan x = 2 cos x 1 ! tan 2 x 2 tan x = 2 cos x 1 ! tan 2 x ) ) sec x 2 sin 2 multiply sin x = simplify No Solution ) factor sin 2x = 0 2x = " n "n x= 2 " 3" x = 0, , " , 2 2 2 sin x ! 1 = 0 2 2 " 3" 5" 7" x= , , , 4 4 4 4 solve ( cos 4 x = cos 2 x 1 + cos 2u cos u = 2 2 1 ! cos 2u tan u = 1 + cos 2u 2 u 1 ! cos 2u =± 2 2 u 1 ! cos 2u sin u tan = = 2 sin u 1 + cos 2u and cos u 2 cos 3" 2 set to zero cos 2x = !1 2x = " + 2" n " + "n 2 " 3" x= , 2 2 x= 2 rewrite (1 + cos 2x ) 2 rewrite 2 22 1 2 = (1 + cos 2x ) 4 1 = 1 + 2 cos 2x + cos 2 2x 4 1! 1 + cos 4x $ = # 1 + 2 cos 2x + 4" 2 %& u 1 + cos 2u =± 2 2 depened on the quadrant in which ) ! 1 + cos 2x $ =# &% 2 " = Half – Angle Formulas sin simplify rewrite Rewrite cos 4 x as a sum of first powers of cosine. 1 ! cos 2u sin u = 2 2 x= Example More Formulas Power – Reducing Formulas rewrite sin x = !1 2 sin 2x ( cos 2x + 1) = 0 cos 2x + 1 = 0 2 sin 2x = 0 2 sin x = ± " 5" , 6 6 1 2 sin 4x + 2 sin 2x = 0 2 sin 2x cos 2x + 2 sin 2x = 0 set to zero cos x 2 sin x ! 1 = 0 factor Try : Solve sin 4x = !2 sin 2x on [ 0, 2" ) rewrite 2 sin 2 x cos x ! cos x = 0 u x= set to zero 2 sin x cos x ( sin x ) = cos x 2 3! ) ( )) x ! sin x ! 1) = 0 rewrite Use sin 2x = 2 sin x cos x rewrite and substitute The signs of sin 2! sec x ( 2 sin x ! 1) ( sin x + 1) = 0 factor sec x = 0 2 sin x ! 1 = 0 sin x + 1 = 0 divide by 2 Try : Solve sin 2x sin x = cos x " 3" , 2 2 ( multiply sin x cos 2 x sin 2 x = ! cos x cos x cos x cos 2 x sin x cos 2 x sin 2 x = ! cos x cos x cos x 2 sin x sin x cos 2 x + ! =0 cos x cos x cos x x= ( ( rewrite tan x = cos x ! cos x tan 2 x cos x = 0 ! 1 sin x + sin 2 x ! cos 2 x = 0 cos x sec x sin x + sin 2 x ! 1 ! sin 2 x = 0 Move 2 cos x and use tan 2x formula ( y = cos 2x -1 Solve tan 2x ! 2 cos x = 0 ( graph this function y = cos x 0 Example tan x = cos x 1 ! tan 2 x 1 2! Period : =! 2 sin x = 2 )) rewrite 2 Use cos 2x = 2 cos x ! 1 rewrite cos x ( 2 sin x ! 1) = 0 factor cos x = 0 2 sin x ! 1 = 0 ( ( ) factor 1 • cos x + cos x ! 1 2 = 2 cos 2 x ! 1 Solve sin 2x ! cos x = 0 " 3" x= , 2 2 )( ( = = 1 ! 2 sin u 2 sin x cos x ! cos x = 0 Half-Angle Formulas 4 cos x ! sin x = cos 2 x + sin 2 x cos 2 x ! sin 2 x 4 2 Use sin 2x = 2 sin x cos x rewrite and substitute Power-Reducing Formulas p1 ( u 2 lies. = ) 1 ! 2 4 cos 2x 1 cos 4x $ + + + 4 #" 2 2 2 2 &% 1 = ( 3 + 4 cos 2x + cos 4x ) 8 power rule rewrite multiply rewrite rewrite rewrite factor solve A2HCh0705 Multiple Angles and Product to Sum Formulas more examples Example Find the exact values of cos 165º . cos 165º is in quadrant II , use negative cos 165º = cos u 2 , so u = 330º 1 + cos 330º 2 cos 165º = ! rewrite 1+ 3 2 =! 2 Product-to-Sum Formulas simplify More Formulas simplify simplify solve solve Sum – to – Product Formulas 1 sin u sin v = "# cos ( u ! v ) ! cos ( u + v ) $% 2 1 cosu cos v = "# cos ( u ! v ) + cos ( u + v ) $% 2 1 sin u cos v = "#sin ( u + v ) + sin ( u ! v ) $% 2 1 cosu sin v = "#sin ( u + v ) ! sin ( u ! v ) $% 2 Example replace by using the formula " 195º !105º % " 195º !105º % sin195º ! sin105º = 2 cos $ '& sin $# '& 2 2 # = 2 cos (150º ) sin ( 45º ) rewrite " 3% " 2 % = 2$ ! '$ 2 ' 2 # &# & simplify =! Example Verify the identity 6 2 simplify sin 7x ! sin 5x = ! cot 6x cos 7x ! cos 5x replace by using the formula " 7x + 5x % " 7x ! 5x % 2 cos $ sin 2 '& $# 2 '& # sin 7x ! sin 5x = cos 7x ! cos 5x " 7x + 5x % " 7x ! 5x % !2 sin $ sin 2 '& $# 2 '& # " 12x % cos $ # 2 '& = " 12x % ! sin $ # 2 '& simplify cos ( 6x ) = ! sin ( 6x ) = ! cot 6x simplify rewrite Try : Verify the identity cos 4 x ! sin 4 x = cos 2x cos 4 x ! sin 4 x = cos 2x 2 2 cos x + sin x cos 2 x ! sin 2 x = factor )( ( ) ) 1 cos 2 x ! sin 2 x = cos 2x = ! u + v$ ! u ' v$ sin u + sin v = 2 sin # cos # " 2 &% " 2 &% ! u + v$ ! u ' v$ sin u ' sin v = 2 cos # sin " 2 &% #" 2 &% ! u + v$ ! u ' v$ cosu + cos v = 2 cos # cos # " 2 &% " 2 &% ! u + v$ ! u ' v$ cosu ' cos v = '2 sin # sin " 2 &% #" 2 &% Example Find the exact value of sin195º ! sin105º ( square / simplify More Formulas Product – to – Sum Formulas Sum-to-Product Formulas # 1 " cos x & 2% = cos x 2 (' $ 1 " cos x = cos x 1 = 2 cos x 1 = cos x 2 ! 5! x= , 3 3 rewrite 2+ 3 2 =! p2 x Solve 2 sin 2 = cos x on [ 0, 2! ) 2 x 2 sin 2 = cos x 2 2 # 1 " cos x & 2% ± replace sin with formula, use ± ( = cos x 2 $ ' Example simplify rewrite / Double Angle Solve cos 3x + cos x = 0 on [ 0, 2! ) cos 3x + cos x = 0 " 3x + x % " 3x ( x % 2 cos $ cos $ =0 ' # 2 & # 2 '& 2 cos ( 2x ) cos ( x ) = 0 replace by using the formula simplify cos ( 2x ) cos ( x ) = 0 simplify cos x = 0 cos 2x = 0 ! ! x = + 2! n 2x = + 2! n 2 2 or ! x = + !n 3! 4 x= + 2! n or 2 3! 2x = + 2! n 2 3! x= + !n ! 3! 4 ! 3! 5! 7! x = 2 , 2 x= , , , 4 4 4 4 solve each Try : Find the exact values of cos120º + cos 30º replace by using the formula ! 120º +30º $ ! 120º '30º $ cos120º + cos 30º = 2 cos # cos # 2 2 " %& " %& ! 150º $ ! 90º $ = 2 cos # cos # " 2 %& " 2 %& simplify = 2 cos ( 45º +30º ) cos ( 45º ) rewrite = 2 cos ( 75º ) cos ( 45º ) simplify = 2 ( cos 45º cos 30º ' sin 45º sin 30º ) cos ( 45º ) rewrite ! 2 3 2 1$ 2 = 2# • ' • • 2 2 2 &% 2 " 2 simplify ! 6 ' 2$ 2 12 ' 4 2 3 ' 2 = 2# = &• 2 = 4 4 4 " % simplify