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Unformatted text preview: MAT 106— 7 _ Instructor: Mike Wang NAME: ﬁnkP— 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. *(True1ftan3 > Oandcscﬂ > 0, thenﬂmustlieinthethirdquadrant. *.r False): cos(y—1I3{— == —cos(—f—“2). *crmeome— = sills) * ' . ‘Falselz' The xintercepts 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. TIEmow 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:
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\_s2—:§__. is. / W Z
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6. (12 pts) beget/Sam“);
®Ve1ify the identity tan .L:.£9§x. (6 pts) ‘ TlTWES
.‘ . (%=~) sinfﬁ \(\_;'7i$_\ (A/DEN
NOTE. \ Cosx __ \ LOSCI 3 ___ SMA L
éWkX ‘ SmO.‘ '5) Znsimémsé;
. (X
: i$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 : upstagewar. 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, ZlTﬂbFTJTewe‘ ><t ‘3‘: macM21 34 a: 349% “£33 “ tau/36m 1‘ [thxﬂZ.~ [."th'lza; 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—ointart)
~ Lc$2tc°stf ~ sKnZtsiw’c § : CLOSz'E’ — $§w&"t\ “St "' (2siw’twsk—§S?«dt
2‘: cos5t —— $1.54: east  ZS‘MLtce'S—t = COSS‘E: " gsié’cwst (g: tQMXZQXi—‘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)
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+ 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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 Spring '08
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