Chapter 6. Inverse Circular Functions and Trigonometric Equations. Section 6.1 Inverse Circular Functions y = 0

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1 Chapter Inverse Circlar Fnctions and Trigonometric Eqations Section. Inverse Circlar Fnctions. onetoone. range. cos... = tan.. Sketch the reflection of the graph of f across the line =. 7. (a) [, ] é ù (b), êë úû (c) increasing (d) is not in the domain 8. (a) [, ] (b) [0, ] (c) decreasing (d) is not in the range 9. (a) (, ) æ ö (b) ç, çè (c) increasing (d) no é ö æ ù 0. (a) (,] È [, ) ;,0 È 0, ê ë çè úû. (b) (,] È [, ); é 0, ö È æ, ù ê ë èç úû (c) (, ); (0, ) cos a. Find tan + ( or 80 ). a. = sin. sin ( ) does not eist does not eist sec 0 csc æ ö ç çè. does not eist. does not eist.. csc æ ç ç ö çè does not eist sin does not eist. 89 Copright 0 Pearson Edcation, Inc.

90 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 8. cos ( ) does not eist 7. 9. 7.78 0..99..00970. 97.707. 0.9879. 9.08..789. 0.77 7. 8.789 8. 0.7 9..9008 0. 0.9098..907..080. 0.8798. 0.7879.

2 90 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 8. cos ( ) does not eist is not in the domain of = sin. (Alternativel, o cold state that.00 is not in the range of = sin. ) 7. In each case the reslt is. The graph is a straight line bisecting qadrants I and III (i.e., the line = ). 77. It is the graph of =. Copright 0 Pearson Edcation, Inc.

3 Section. Inverse Circlar Fnctions It does not agree becase the range of the æ ö inverse tangent fnction is ç,, çè not (, ), as was the case in Eercise sin θ = Then, (a) (b) θ =. (a) 8 (b) 8 (c) Copright 0 Pearson Edcation, Inc.

4 9 Chapter Inverse Circlar Fnctions and Trigonometric Eqations. + (d) tan ( θ + α) = = and tanα = tanθ + tanα tan( θ + α) = tanθ tanα tanθ + = æö tanθ ç çè tanθ + = tanθ ( tanθ) = ( tanθ + ) tanθ = tanθ + = tanθ + tanθ = tanθ ( + ) tanθ = + æ ö θ = tan ç çè + (e). m. (Note: De to the comptational rotine, there ma be a discrepanc in the last few decimal places.) (f) æ 7 ö æ0ö θ = arctan arctan ç è00 çè..7%. cbic ft Section. Trigonometric Eqations I. Solve the linear eqation for cot.. Solve the linear eqation for sin.. Solve the qadratic eqation for sec b factoring.. Solve the qadratic eqation for cos b the factoring.. Solve the qadratic eqation for sin sing the qadratic formla.. Solve the qadratic for tan sing the qadratic formla. 7. Use the identit to rewrite as an eqation with one trigonometric fnction. 8. Use an identit to rewrite as an eqation with one trigonometric fnction. 9. 0º is not in the interval [ 0,0 ). 0. To show that 0,, is not the correct î soltion set to the eqation sin = cos, show that at least one element of the set is not a soltion. Check = 0. sin = cos sin 0 = cos 0? 0=? 0= 0 Tre = 0 is a soltion. Check =. sin = cos sin = cos? = 0? = Tre = is a soltion. Check =. sin = cos sin = cos? = 0? = False = is not a soltion. Copright 0 Pearson Edcation, Inc.

5 Section.: Trigonometric Eqations I 9... In general, when o sqare both sides of an eqation or raise both sides of an eqation to an even power, o mst check all soltions in order to eliminate an etraneos soltions. 7, î î, î., î. Æ. Æ 7.,,, î 7 8.,,, î 9. { 0.,,, î. 7,, î. 0,, î. {0, 0, 0, 00. {0,,. {90, 0, 0. { 0,, 0, 7. {,,, 8. { 0, {, 0. { 90, 70. { 0, 0, 0, 80. { 0, 90, 80, 70. { 0,,, 80,,. {,,,. {.,., 87.9,.. { 78.0, { 0., 9., 8., { 8., 0.8, 8., Æ 0. Æ. {7.7, 9.. {.,.7. { n, where n is an integer. { + 80 n, where n is an integer., + n + n, where n is an integer î. + n, where n is an integer î 7. { n, n, n, n, where n is an integer 8. { n, n, n, where n is an integer 9. + n, + n, î and + n, where n is an integer n and + n, where n is an î integer. {80 n, where n is an integer. { n, where n is an integer. {0.87+ n,.+ n,.908+ n, and.80 + n, where n is an integer. is {.88 + n,.77 + n, where n is an integer. {. + 0 n and. + 0 n, where n is an integer. { n, + 80 n, where n is an integer Copright 0 Pearson Edcation, Inc.

6 9 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 7. { + 80 n and n, where n is an integer 8. { n, + 80 n, where n is an integer 9. {0.80,.9 0. {0, (a) 0.00 and (b) [0.00, 0.00]. (c) otward.. If cot csc = 0 has no soltions in the 0,, then the graph of interval [ ) = cot csc will have no intercepts in this same interval.. (a) sec (b) sec (c) 0. sec. (a) sec (b) sec. (a) One sch vale is (b) One sch vale is. In the second line of the soltion, both sides of the eqation were divided b sin. Instead of dividing b sin, one shold have factored sin from sin sin. In the process of dividing both sides b sin, the soltions of = 0 and = were eliminated. Section...,, î,, î8. { 0, 0, 0, 0. {, 80,, 70. Trigonometric Eqations II tan θ ¹ tanθ for all vales of θ ,,,,,, î 9. {90, 0, 0 0. {0, 0, 0, 80, 0, ,,,,, î ,,,,, î {7.,., 7., 9.. {, 7, 9, Æ 0. Æ, î 7 0,,,,,,, î 0,,, î 0,, î. {80. {0 Copright 0 Pearson Edcation, Inc.

7 Section.: Trigonometric Eqations II 9.,, î 7.,,, î. n n +, +, where n is an integer î 7. + n, + n, where n is an integer î 7. {70 n, where n is an integer 8. { n, where n is an integer n, + n, where î n is aninteger 7 + n, + n, where n is an î integer. {0 + 0 n, n, n, where n is an integer. {0 + 0 n, n, n, where n is an integer. n, + n, + n, where î n is an integer. + n, + n, + n, + n, î where n is an integer or + n, + n, where n is an integer î. {.8+ n,.9+ n, where n is an integer. {.0 + n,.0 + n, where n is an integer 7. { n, n, n, n, where n is an integer or { n, n, where n is an integer 8. {. + 0 n,. + 0 n, n, n, where n is an integer or { n, n, where n is an integer 9. {0 + 0 n, n, n, n, n, n, where n is an integer or { n, n, n, where n is an integer 0. {0 + 0 n, n, n, n, where n is an integer or {80 n, n, n, where n is an integer. î. {0,. {0.,.0. {.89,.0. {.80. {0.99, (a) (b) The graph is periodic, and the wave has jagged sqare tops and bottoms. (c) This occrs for the time intervals (0.00, 0.009), (0.0, 0.08), (0.07, 0.07). Copright 0 Pearson Edcation, Inc.

9 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 8.

(a) (b) beats per sec (c) The nmber of beats is eqal to the absolte vale of the

. (a) The average monthl temperatre is F in the seventh month, Jl.

eleventh month, November.. (a) The average monthl temperatre is 70.

(b) The average monthl temperatre is F when =. (dring Febrar) and when =.

00079, 0.00989, 0.089, 0.080 (c) 0 Hz (d). (a) (b) (c). (a).89 (b) 7.08 0. (a) = 9.

8 9 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 8. (a) beats per sec 9. (a) (b) beats per sec (c) The nmber of beats is eqal to the absolte vale of the difference in the freqencies of the two tones.. (a) The average monthl temperatre is F in the seventh month, Jl. (b) The average monthl temperatre is 9 F in the second month, Febrar, and in the eleventh month, November.. (a) The average monthl temperatre is 70. F when = (dring April) and when = 0 (dring October). (b) The average monthl temperatre is F when =. (dring Febrar) and when =.8 (dring November) sec sec. 0.00sec sec Chapter Qiz (Sections..). [, ]; range: [ 0, ] (b) , , 0.089, (c) 0 Hz (d). (a) (b) (c). (a).89 (b) (a) = 9. means abot 9. das after March, on Jne 0. (b) = 7.7 means abot 7.8 das after March, on December 9. (c) = 8.8 means abot 8.8 das after March, on Febrar.. (a) (b). {0, 0. {0, 80, {0.089,.,.70, ,,, î n, + n î Copright 0 Pearson Edcation, Inc.

9 Section.: Eqations Involving Inverse Trigonometric Fnctions (a) 0sec (b) 0.0sec Section. Eqations Involving Inverse Trigonometric Fnctions. C. A. C. C. arccos =. arcsin 7. = arccot 8. = arcsec 9. = arctan 0. arcsin =. arccos =. = arcsin( ). æ ö = arccos ç çè. = arccot. = + arccos. = ( + arctan ) 7. = arcsin( + ) 8. = arccot( ) æ + ö = arcsin ç çè æ ö = arccos ç çè sec æ = ö ç çè. csc æ + = ö ç çè. First, sin sin( ).... ¹ If o think of the graph of = sin, this represents the graph of f ( ) = sin, shifted nits down. If o think of the graph of = sin( ), this represents the graph of f ( ) = sin, shifted nits right. cos doesn t eist since there is no vale sch that cos =. î î 7. { 8. Æ 9. { î î î î. î. { 0. î 7. î 8. Æ 9. î Copright 0 Pearson Edcation, Inc.

8) (b) For = t, Pt ( ) =.000sin(0 t+.8) Pt ( ) + Pt ( ) =.00sin(0 t+.0) +.00sin( t +.) The two graphs are the same. + 9.

10 98 Chapter Inverse Circlar Fnctions and Trigonometric Eqations 0. î. { 0. {0. Y = arcsin X arccos X 8. (a) φ» 0.70; f = 00, P= 0.00sin( 00 t+ 0.7) (b) For = t, Pt ( ) =.00sin(00 t+.7) æ ö Pt () + Pt () =.00sin ç 00 t+ + çè 7 æ ö.00sin ç 00 t + çè. Y = arcsin X arccos XY =. {.. {.8 7. (a) A».000; φ» 0.8; P= 0.000sin( 0 t+ 0.8) (b) For = t, Pt ( ) =.000sin(0 t+.8) Pt ( ) + Pt ( ) =.00sin(0 t+.0) +.00sin( t +.) The two graphs are the same (a) tanα = and tan β = z z + (b) = tanα tan β (c) (d) æ tan arctan β ö α = ç è + æ( )tan arctan + α ö β = ç çè p p 0. (a) = sin, (b) The two graphs are the same. (c) (d) tan = æ ö = arctan ç çè E. (a) t = arcsin f E (b) sec ma Copright 0 Pearson Edcation, Inc.

é ù [, ];, êë úû (b) (i). ft or 0.9 ft (ii). ft or 0.0 ft (c) (i) 0. (ii) 0. [, ]; [ 0, ].

11 Chapter : Review Eercises 99. (a) θ = α β Since tanα = α = tan ç æ ö çè and tan β = β = tan æ ç ö, çè we have æö æö θ = αβ θ = tan tan ç è çè Chapter Review Eercises. é ù [, ];, êë úû (b) (i). ft or 0.9 ft (ii). ft or 0.0 ft (c) (i) 0. (ii) 0. [, ]; [ 0, ].. t = sin t t (a) = sin = arcsin t = arcsin t = arcsin (b) If = 0. radian, t = arcsin 0.9 t» 0.7sec. é ù [, ];, êë úû. False. The range of the inverse tangent æ ö fnction is ç, çè, while the range of the inverse cotangent is ( 0, ).. False. æ ö arcsin ç =, not çè. Tre Copright 0 Pearson Edcation, Inc.

12 00 Chapter Inverse Circlar Fnctions and Trigonometric Eqations θ = θ». 9. θ» θ».. θ» 7.0. θ» , î 7. {0., ,,.9 î ,,, î 7 9,,,,,,, î ,,,,,,, î { 0+ n, where n is an integer or { n, where n is an integer.. + n, + n, + n, î where n is an integer 7 + n, + n, + n, + n, î where n is an integer or + n, + n, where n is an integer î. {70. {,.,,. 7. {, 90,, {, 7, 9, 9. {70., 80, {., 8.,., 98.. { n, n. {0 + 0 n, n, n. { n. {. Æ. 7 î Copright 0 Pearson Edcation, Inc.

Chapter : Test 0 7. î 8. = arccos 9. = sin ( ) 0.

arccos ç çè 0. (a) Let α be the angle to the left of θ. c.

8. the light beam is completel nderwater. (a).

= csc Ths, we have + 0 tan( α + θ) = æö α + θ = arctan ç

13 Chapter : Test 0 7. î 8. = arccos 9. = sin ( ) 0... æ + ö = arcsin ç çè = æ arctan ö ç çè 0 æd 0ö t = arccos ç çè 0. (a) Let α be the angle to the left of θ. c. sinθ = c sinθ c sinθ sinθ =.7 = c sinθ sinθ 8.8. the light beam is completel nderwater. (a). (b) 90 (c) = csc Ths, we have + 0 tan( α + θ) = æö α + θ = arctan ç çè æö θ = arctanç α çè æö æö θ = arctan arctan ç è çè (b) The maimm occrs at approimatel 8.0 ft. There ma be a discrepanc in the final digits. 8. (a), (b)in both cases, Chapter Test. sin 0.» 0.8 é ù ;, êë úû [, ] Copright 0 Pearson Edcation, Inc.

14 0 Chapter Inverse Circlar Fnctions and Trigonometric Eqations. (a) (b) (c) 0 (d). (a) 0 (b) (c) (d) 0. (a). (b).7 (c) n, where n is an integer î 8. (a) (b) 9. (a) 0. (b) = arccos æ ö = cot ç çè î î 7 sec, sec, sec. (a) (b) 9. Since sinθ, there is no vale of θ for which sinθ =. Ths, sin is not defined. æ 7. arcsin sin ö æ ö = arcsin = ¹ ç è èç {0, 0 0. {90, 70. {8.,, 98.,. 0,, î 7 7.,,,,, î. {0.9,.09,.0,.7. {90º + 80ºn, where n is an integer.. + n, + n, î where n is an integer Copright 0 Pearson Edcation, Inc.

Differentiation 9F. tan 3x. Using the result. The first term is a product with. 3sec 3. 2 x and sec x. Using the product rule for the first term: then

Differentiation 9F. tan 3x. Using the result. The first term is a product with. 3sec 3. 2 x and sec x. Using the product rule for the first term: then Differentiation 9F a tan Using the reslt tan k k sec k sec 4tan Let tan ; then 4 sec and sec tan sec d tan tan The first term is a proct with and v tan and sec Using the proct rle for the first term: sec