MAT106-WINTER08-MIDTERM-SOLUTIONS - MAT 106— 7 Instructor...

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Unformatted text preview: MAT 106— 7 _ Instructor: Mike Wang NAME: fink-P— Ka‘f Midterm Show all your work clearly! If you finish early, don’t forget to check your work. 1. Answer the following: (10 pts) (1 pt each) @r False): a = g— rad and p = 3% rad are coterminal. ” (T rue 0. An acute angle has angle measure between 90 and 180 degrees. *(True-1ftan3 > Oandcscfl > 0, thenflmustlieinthethirdquadrant. *.r False): cos(y—1I3{— == —cos(—f—“2-). *crmeome— = sills)- * ' . ‘Falselz' The x-intercepts for y = tanx is x = 15 + kw: such that Iris aninteger. {EB . 2 *@or False): Assuming both sides are defined, tanx = tan(x+k1r) such that k is an integer for all values of x. TIE-mow Ifcosx = 0, thenx = knsuchthatkisaninteger. a‘(Tr'ueo w eperiod fory= cscxisu. * (True [email protected] domain of y = tanx is all real numbers except [at such that k is an integer. , I 2. Find the exact value of the following expressions. Be sure to sketch a reference triangle for each. (9 pts) ' (i)cos(?T”) : ‘d’li (ii)tan(—-§-) :3}: - | (iii) 511101.?) : 12: 3. Find the exact value of each of the six trigonometric functions if the terminal side of 0 contains the point P(—3,—4). (6 pts) ‘ 4. Find the least positive 9 in degree and radian measure for tang = ——1—. (8 pts) J? L MUST 32 Gwen: e: @646” =|§d M —r3 9 _ F W _ SW 5. Find the exact value of cos(2a) given that sin a = 187 and a is in quadrant II. (9 pts) E: —' ., ,‘ 2 N3; gosczoq - \ Lego: 7' l ‘ 1(v7 (:4 :x—ztzaa -\_s2—:§__. is. / W Z I $\N 6. (12 pts) beget/Sam“); ®Ve1ify the identity tan .L:.£9§-x-. (6 pts) ‘ TlTWES .‘ .- (%=~) sinffi \-(\_-;'7-i$_\ (A/DEN NOTE. \ Cosx __ \ LOSCI 3 ___ SMA L éWkX ‘ SmO.‘ '5) Znsimémsé; . (X : i$kvx 'i 1; ELC’OS‘E‘i _ gm _ 25. C05 tM 2‘ (ii) Using the identity in (i), find the exact value of tan(15° ). pts) {'73 3—0- .— \ “ C9330 \ - ’2': NUTE‘ W\§° 1: WL 2_>° ' ': ,, , ' $\n?>o° 3‘ UES m - QUAD: ._ 1 ’- 16 ‘ 4'5 7. Two buildings are 300 ft apart. From the top of the shorter building the angle of elevation of the top of the taller building is 23 degrees, and the angle of depression of the base of thetallerbuflding is Sfifldegrees (see figure). How tall is each building? (12 pts) 71%;; {ET1 X = Ham—w or- gans—Tea BVHJMMA i "3 t3 : upstage-war. w uam—W Semeaqswas KHz, (on KW}, = Hewitt 0? Mwakgwmwa} I<—-—————-'300 I‘t—'——-—-—>¥ Note: EMF; 1: “V: -_—> (3-: 2soo’cmzf'xn134q X o (Wil- .= n, we! ( am$5= $00 “=3 X Boowsép A, ZlTflbF-TJTewe‘ ><t ‘3‘: mac-M21 34 a: 349% “£33 “ tau/36m 1‘ [thxflZ-.~ [."th'lz-a; W'LX o/ 8. Ven'fy the following identities. (18 pts) (i) tan2(—x) — Egg—fl = seczx (6 pts) ._ 72 Hut-)0 1 35:5:- sf“): 1 = tom X ‘F \ -= See—1X , TM! (\ENT / 5m \DEN‘UT’T -- Ztanx _ (11) m — 13112.76 (6 ptS) N§= {Mix -- ’cmLM x3 ' H E 2; § X ZWX \ 1* tmzy C133 \NE a/ sou \DENTITT (iii) cos 3t = cos3t — 3 sinztcost (6 pts) NOTE CosLBtW = C—oin-tart) ~ Lc$2tc°stf ~ sKnZtsiw’c § : CLOSz'E’ — $§w&"t\ “St "' (2siw’twsk—§S?«dt 2‘: cos-5t —— $1.54: east - ZS‘ML-tce'S—t = COSS‘E: " gsié’cwst (g: tQMXZQX-i—‘T/fll NE A:\/B:2,;C='“/2_/D:O 9. Graph y = tan(2x + 7:) for two cycles. Be sure to state the amplitude, period, range and equations of asymptotes. Show work. (12 pts) — W L UD ‘3 x: J74 UD BI — E V 6 ‘ z WQE: C'— 09, 6°) ’< Pom 661‘- + gk =r JrL NF Extra Credit (5 pts) Verify seczx~csc::t:xcsczxcoszx : tan4x. L 1 Z Z. Non-:2 $QLX“CSCX+CSCXCDSX Z M 1.. * SEC X " \ Co,” X ': {2. ._. $9.ch — (LSQZX U - mix) COL X ‘ L - (3‘ ~ oscax vaé‘x 0’" X — CDth = 'Ecuf x V l L L / a/L : §Qc X S.‘ X \wx ' @th ...
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