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Completed_5.2S16.pdf

# Completed_5.2S16.pdf - Section 5.2 Trigonometric Equations...

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Unformatted text preview: Section 5.2 Trigonometric Equations A. Trig Equations in the Form a sin(x — c) = k, a cos(x — c) 2 k, and a tan(x — c) =2 k Ex. 1: Find all solutions of each equation. Express all solutions 1n radians. ‘5 \3”\{2:‘2€“7‘2‘723\472\2:223 2 722:} 222;" 3' Si” :77 "' 32222 72:2 ‘22 2:7: 2 43722; C373: ’ >4 7 3:3? 23:: “2‘2: (“2:223:77 ”“213 22% K3: :7 (7,72% 2 iii KL. "Z; L2 232775723 (:12: :32 bL" (7712’ 2%“V« 1'2 22 7 7? 2:74.: 3%: :7 222222\ b cosx : 15 Wm M 7 m, M y 04773 2 < (:7 22": (WIDE (23:2 ‘2. I; 2“/2: 2:2 C233 4232222223 >4 ‘3 7779 @773 ‘- "W “ 727727, 3:72:22 QT?” NT “222 ”YT/((1:17 WWW/[A X 2%: (2277“ (3‘7 ><777W 3253232 273222232 2223:2718 2 »:_ <23 W 37337 @337; 2243272. x 4“ 222% wheﬁ 22 ”i 777737 2222:4222 22 2: 2322223 €222 {Wﬂi‘m :23 4:229: 22222:: @233; 273222223223 2‘29 22?: R ”We; Lei >43 mill/L3 W6C“? 2‘ 33% E3 “:X % 33’ £43 ”3:7 3% (3% 3 3*; am (“‘3 3333 Tm (3353:3333: <3: 23mg - 3/»; {met/3 (*3 3 ”*3/49 5333'. 33633 W: Mr: 9: TP’YF/épt 5% ”w w 331%“ 9:: i @ ”if” 3‘ @333" 5:: 3; :3 [@133 EX. 3: Find all solutions in the interval [0,211) of the equation “' *, 2 sin (x — E) + 1 = 2 3533331533 33W ii 9i *3 25 x 3 5T: ”73‘” ; \O’TY 3W .3 :3le 3 (.3, 3 “a?” a ”'33” X W * “T.” W "i 3 3 I 33, \g/iﬂ a? (YT/953 m ill/g?” +% H AZ (3 5,; V2: l2 if; 733 328% @132 \$5 B. Equations Involvmg Multiple Angles W3 63 C136“: \$333663 LE3“ Q) g: 3343 r 7 3. ’ 25:; i. 1?? Ex 4: Find all Soluticms Of the equation COS 3x = gin the interval [0,27t). “WW ‘7 {33%) Q I/Z ‘ a] Q13 @3323 Wme Q : 3:” Qaa 3% 3»sz «376%, 3% 3 513333 233' 3 3333/5 ©133C©E® :3 V3 tom/363 33 93; ”33323“ 3 333373 (93 3 MTV/332T? 3 3737/3 Q \$233-«Ezigi ' 3% FT 3-: \WT Xi '3’ “/3735" ”Cl 3&3 kl 3 r3 3: 3133: 33: 33:”: W” Xﬁm/q 35 igiq gqlgmgqlawéfwaf 3, - 34 “is???“ x2} EWWQ 3 /q WHKW 533:3», mumymwwygwmw—w33333.,er Mme/WW 3.3333 ,, b 2‘: i/ZJ WWW-“"7 ‘3 CD?“ (:3 :ﬂlmﬁ"”3\$ %ﬂ€ﬁ 9‘5le Ex. 5: Find all solutions of the equation sing = %/—E in the interval [0,210. g .33. Z 3R5, 3% mi 33/22 ®\ﬂ@<:® m Qm‘ (DE 9V: 33%;!3 @2 3 53/3 .35, :: Lviﬂiffﬁ X m; STWE pm 23 73, ”if” g Q 3 3.3; e 3533/3 M7 mm 03333333 3333333 M i " \ New CSXM C. Trigonometric Equations and the Zero-Product Property Ex. 6: Find all solutions of the equation (2 sinx —— x/s) (7 tan x + 2)— — 0 in the interval [0, 2n). Round all solutions to multiples of 1: or four decimal places. (2: QEQEEE J2EE EQEEEE E23 E53 Z QEQ EEY‘Z “ 53 E EQEEE: E Z: , EEEEEjﬁy/EE EE @7933 3W“)? :( M f/Z ‘4?)quwa X: m 21!?“ij ‘E’Q ll) CE”: (EX Q3; 33:): “(#2 x >< : TEE/L,t E KEV/pk (33,: >42: EEE EQEEE ﬂkvaw ”ix ZEEE EQEE XE; E ? ”KT/w Eﬂ/E 2E 5599M {a (“363‘ 63:; i E EX. 7: Find all solutions of the equation 2 sin2 9 — 5 sin 9 + 2 = 0. Express the solutions in radians. Zawxqgw Same; EELECE W: 3mg E @E E TE: E. QEEEEE QEEE’E E in + Wm ‘ E D. Equations with More Than One Trigonometric Function Ex. 8: Find all solutions of the equation 2 sin2 6 + \/§ cos 9 + 1:0 in the interval [0.212) 222" 2 22 21535429 33332333 @2223“ 22:3: 29 2 2 B2 223323 22%; 2223? Kb”: 2:9 23 2:22:33 Ogﬁ‘C32": CE“ “Z 33 5 3 f“ 3 232C337 VW "5 ’2” ’9’ 5 3’ ‘ 9 2:222 292322222 3Z2? 5, Z; :1: (,0 W323 } E (322:1 "f; E. Extraneous Solutions EX. 9. Find all solutions of the equation 2/? cos x— — sin x + 1' 1n the interval [0,210. Wnoﬁzz: :322’3 22: 22223 (W5 C239 241.32 ‘2? (611:2 x 22mg 9 C393? 22 “22 (92:22 223(3er 2 3,33) :(2 2 321:2? 22:“) ““ 1:222:23 22 2 9 2:32:22 2 2 3 22 3:321:22: 22’ ‘2': 3322:2123 2 23:21:22: 222 253219 / ¥_ 22313222952 5. £22 2.2 3219222 22:22 2:222; :22 '23; (:13:ch ' , 3 1 32:75 CQEZQ’E v32 223323 212‘, 33,2 Q: 229222 23 2323332322233ng 2320: “2:2; {5%(23333 1;: 2’22’22’1 \/ C) /3m?“><22:‘22::22<333 3 2: 2 33 22 “3:222:21 22 3 Z23 <23"ng 1:219:22 212 22 ’3 CC): 2532‘?) 2% ’ 33507 X “’23 :2 ' 2 m 1‘2 w ‘ :3 33’... 313 22 CD 322122222233 {30:331233331 Qa’mKW/ZZ €‘1Vﬁﬁﬁmk 1 QT?“ 333:: - 2 x: 522 "z Reo:{;§}):32929922 (9 ’ €10 ' Z2” ...
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