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A2HCh0606 Inverse Trigonometric Functions Homework and Reading Read p4505–510 (1) HW p511 #1– 47 odd, skip 17 (2) HW p512 #49–73 odd. 91, 97, skip 63 Goal Inverse Sine Function p1 Students evaluate and graph the inverse sine and other trigonometric functions, and evaluate and graph the compositions of trigonometric functions. Inverse Sine Function Def. of Inverse Sine Function The inverse sine function is defined by y = sin x Horizontal Line Test 1 ! ! ! ! -! ! " 2 ! 2 0 ( ! ! 2! where ! 1 " x " 1 and !# 2 " y " # 2 . The domain of y = arcsin x is [ !1, 1] , and the range is $% ! # 2 , # 2 &' . ! # " # $ -1 If we restrict the domain to this interval, we have an inverse to sin x Example A The following holds true for the interval #$ !" 2 , " 2 %& sin !1 x = y iff sin y = x II y = sin x takes on its full range of values. ! 1 ' sin x ' 1 III y = sin x is one – to – one. Example B Find sin !1 (1) ( Find the arcsine of 1) sin !1 (1) = y " sin y = 1 sin # 2 = 1 y = sin x is increasing. I ) y = arcsin x y = sin !1 x if and only if sin y = x ( ) Find arcsin 1 2 arcsin x = y iff sin y = x ( ) arcsin 1 2 = y ! sin y = 1 2 sin " 6 = 1 2 see p474 common angles Try : Find the following B 4 or 45º # " 4 or # 45º " or 180º #1 C Other Inverse Trigonometric Functions " sin #1 2 2 arcsin $ # 2 2 & % ' A sin 0 Other Inverse Trigonometric Functions Example A Def. of the Inverse Trigonometric Functions Domain Range y = arcsin x iff sin y = x Function !1" x "1 y = arccos x iff cos y = x !1" x "1 !# 2 " y "# 2 y = arctan x iff tan y = x !$"x"$ π cos !1 3 2 = y " cos y = 3 2 cos # 6 = 3 2 see p474 common angles 0" y"# ! # 2 < y2π< # 2 Try : Find the following ! 2 π π A ! 2 -1 0 ! 1 -4 " 2 -3 -2 -1 0 ! -1 0 y = arccos x y = arcsin x Using a Calculator 1 " 2 Use a calculator to approximate the values A sin !1 0.5524 (use radians) 2nd sin !1 .5524 enter sin !1 0.5524 " 0.5852 arccos 0.5 (use degrees) 2nd cos !1 .5 enter cos !1 0.5 = 60º Try : Find the following A B C tan !1 3 3 (deg ) # % 3 arccos ! 2 & (rad) $ arctan 0 (deg ) 30º about 2.617 180º 1 -2π 2 # tan !1 1 ( arccos ! 1 2 3 4 C arctan 0 B -π y = arctan x Example B Find cos !1 3 2 cos !1 x = y iff cos y = x ) 4 or 45º 5# or 120º 6 # or 0 or 180º A2HCh0606 Inverse Trigonometric Functions Composition of Functions Composition of Functions Evaluate each A sin ( arcsin 0.12 ) QR : §2.7 For all x in the domain of f and f !1 , inverse function have the properties : f f !1 ( x ) = x and f !1 ( f ( x )) = x ( if ! 1 " x " 1, sin ( arcsin x ) = x ) if ! 1 " 0.12 " 1, sin ( arcsin 0.12 ) = 0.12 Inverse Properties of Trigonometric Functions If ! 1 " x " 1 and ! # " y " # , then 2 2 sin ( arcsin x ) = x and arcsin ( sin y ) = y ( arctan tan 5# 6 B ( ( )) If x is a real number and ! # 2 " y " # 2 , then tan ( arctan x ) = x and arctan ( tan y ) = y ( ) if ! # 2 " y " # 2 , arctan ( tan y ) = y # < 5# , Find the coterminal, 5# ! # = ! # 2 6 6 6 If ! 1 " x " 1 and 0 " y " # , then cos ( arccos x ) = x and arccos ( cos y ) = y Try : Evaluate each A arccos cos 7! 2 p2 Example arctan tan ! # 6 = ! # 6 ) if 0 " y " ! , arccos ( cos y ) = y ! < 7! 2 , Find the coterminal, 7! 2 # 3! = ! 2 ( ) arccos cos ! 2 = ! 2 B sin ( arcsin ( #0.2 )) if # 1 " x " 1, sin ( arcsin x ) = x if # 1 " #0.2 " 1, sin ( arcsin ( #0.2 )) = #.2 Evaluating Composition of Functions Example Example Evaluating Composition of Functions A Find the exact value of cos sin !1 3 5 sin !1 x = y iff sin y = x opp sin !1 3 5 = y " sin y = 3 5 = hyp let u = sin !1 3 5 , find cos ( u ) adj Fact cos = , use a triangle. hyp B ( ) ( y 5 a2 + b2 = c2 a = c !b 2 2 ( ( )) Find the exact value of sin tan !1 ! 1 2 tan !1 x = y iff tan y = x opp tan !1 ! 1 2 = y " tan y = ! 1 2 = adj let u = tan !1 ! 1 2 , find sin ( u ) opp Fact sin = , use a triangle. hyp ! a 3 Example sin ( u ) = ! y c 3x x 1 c c = 9x + 1 4$ ! Find the exact value of sec # arcsin & 5% " arcsin x = y iff sin y = x opp arcsin 4 5 = y ' sin y = 4 5 = hyp let u = arcsin 4 5 , find sec ( u ) hyp Fact. sec ( = , use a triangle adj a2 + b2 = c2 y 5 ! a 4 x a 2 = 25 ) 16 = 9 a=3 5 sec ( u ) = 3 Try : Find sin cos !1 x Try : 5% " Find the exact value of csc $ arctan! ' 12 & # arctan x = y iff tan y = x ( ( ) cos !1 x = y iff cos y = x x adj cos !1 x = y " cos y = = 1 hyp ) opp arctan ! 5 12 = y ( tan y = ! 5 12 = adj let u = arctan ! 5 12 , find csc ( u ) hyp Fact. csc ) = , use a triangle opp ) 1 5 a2 = c2 ) b2 c = 9x 2 + 1 3x sin ( u ) = 9x 2 + 1 c 2 = 144 + 25 = 169 c = 13 13 csc ( u ) = ! 5 !1 Try : ! 2 c2 = a2 + b2 x 2 ! c2 = 5 a2 + b2 = c2 ) ( y c= 5 ( 3x )2 + 12 = c 2 ( ) ) 12 + 2 2 = c 2 2 Find the exact value of sin ( arctan 3x ) arctan x = y iff tan y = x 3x opp arctan 3x = y ! tan y = = 1 adj let u = arctan 3x, find sin ( u ) opp Fact sin = , use a triangle. hyp 2 ( ( a2 + b2 = c2 x a 2 = 25 ! 9 = 16 a=4 4 cos ( u ) = 5 C ) let u = cos !1 x, find sin ( u ) opp Fact. sin # = , use a triangle hyp y x 12 ! c !5 y 1 c2 = a2 + b2 12 = a 2 + x 2 a2 = 1 ! x 2 a = 1 ! x2 sin ( u ) = 1 ! x2 = 1 ! x2 1 ! a x x