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Unformatted text preview: Math 113, EXAM III, April 23rd, 2003. (1 hour) SECTION: Instructor: ‘1:
’50 To receive credit for an answer, you MUST show work justifying that answer.
FIGURES MUST BE CLEAN AND CLEAR. I. (20+10 points)
1) In the (as, y) plane, let V and W be the vectors given by their components: —o —a V =<2,4>, W =<5,t>, where t is an unknown parameter. Show the vector 17 on the ﬁgure. Evaluate its length Evaluate the dot product 17 W
(in terms of t). Determine t so that the vectors V and W are perpendicular. Show the
corresponding vector W on the ﬁgure. 2) Determine the EXACT value of the cosine of the angle 6 between the vector l7 (above)
and the vector T =< 4, 2 >. II. (30 points)
1) Can a triangle have sides of lengths 4, 4 and 7 (units)?
2) Can a triangle have sides of lengths 4, 4 and 10‘? In each case, if it is possible to have such a triangle, draw a ﬁgure and evaluate the angles. Otherwise explain why there is not such a triangle. (Separate Very clearly your answers to
questions 1 and 2.) III. Determine the angle a if a force of 45 lbs is needed to prevent the 70 lbs bowl from
running down the slope. (20 points) “103B IV. . (20+10 points)
1) On the ﬁgure show the vectors V :< 6cos 65°, 65in 650 > ,and W :< 0, —3 >. Evaluate the vector 2 = l7 + W, and show it on the ﬁgure. What is the angle between
the vector V and the positive :2 axis (exact value)? What is the angle between the vector 2 and the positive 3: axis (approximate value)? ‘1 2) A boat whose speed (relative to water) is 6 mi/ h maintains a compass heading of 25°
(i.e. 250 east of north), on a river ﬂowing south with a current of 3 mi/h. In which
direction is the boat actually going? (Although you can treat this question independently,
you can also use the result of the previous question.) V. (10+20 points)
1) Which one of the following formulas is correct? For the one that is correct write a proof in the space provided below. Disprove the other ones by taking simple values of 9:. Write a simple (counter)example on the same line as each wrong formula. (Finding the wrong formulas will therefore allow you to treat question 2, even if you cannot give the proof of the correct formula.) sinm—l—cosa: : ﬁcos(2$)
sinxlcosac = x/isin(x+%)
sinm+coscc = \/§COS(:c+745) ll sin a: + cos a: sin(2m) Proof: 2) Using the answer to question 1), solve sinxl—cosr = with OSmﬁZn. 1
ﬂ: Keep exact values. Math 113, EXAM III, April 23rd, 2003. (1 hour) II. (30 points) 1) Can a triangle have sides of lengths 4, 4 and 7 (units)?
2) Can a triangle have sides of lengths 4, 4 and 10? In each case, if it is possible to have such a triangle, draw a ﬁgure and evaluate the angles.
Otherwise explain why there is not such a triangle. (Separate very clearly your answers to
questions 1 and 2.) 9—) T.“ «LVHMQ/L hQanoclk Ag. 13(1,’ cavwwv emﬂk gm
ebb Qwalﬁo dad/M iodnu’u'oba. BM lo 74H, , Ac thew w “a
\“vx'amxﬁf Wil9" H1» 0% M3415» Alum. 1  q 00+“) Points) \) own». a. M a «tow an MW
1) In the (z, 3;) plane. let V and W be the vectors given by their components: _
a ,4 L. ugh ﬁlm A ammo
V=<2,4>,W=<5,t>, A i Y .1; Lc+h1_g_u,x4)md where t is an unknown parameter. ‘7 IV 06‘", C” at —_ _.';1Z I ’80 CL ’2. l 2 7 0 Show the vector \7 on the figure. Evaluate its length Evaluate the dot product 17  W
(in terms of t). Determine t so that the vectors l7 and W are perpendicular. Show the ~ __
Ive Meodelmwtalm P"? .
""" l 8 o ' — 01 2 2 ﬂ corresponding vector W on the ﬁgure.
me: New —. owl/g. so 590? — L ? ‘ IL 'L ': 2)) )
lVl : l Hf 2 d2 0 L )
III. Determine the angle a if a force of 45 lbs is needed to prevent the 70 lbs bowl from —.—4 _, \7 running down the slope. (20 points)
9L —) . V 3 0
V W. 4 o _, _5 10m A (ﬂaw FV%MV‘£ wt»; mat/ad
A. e “h t :‘ _'T+_ " _2__ MHMW EXP—Q». wail5m.
[the 70m” $6“: (wa’r‘P/W‘) 2) Determine the EXACT value of the cosine of the angle Qbetwee the vector 17 (above) andthevecwrf=<4y2>> m : mm: 1o (0 hmvmd m an AM A, a
, a: m W‘l a» 9 “A Vii". mum): l6. )om MWCJ Mu MW“me
V. 1 ’ ‘6 g \DVOMacl/‘lw\ am} a Brno. alw‘clhkggdpe] '4 — —— :0. .
Sc (me’ Eﬁo £0 wg were) h3=705ulol. go Quickie: 0),: [loo IV. (20+10 points) V. (10+2O points) 1) On the ﬁgure show the vectors l7 =< 6cos 65°, 6sin 65" > ,and W =< O, —3 >. 1) Which one of the following formulas is correct? For the one that is correct write a
Evaluate the vector Z = I7 + W, and show it on the ﬁgure. What is the angle between proof in the space provided below. Disprove the other ones by taking simple values of I.
the vector V and the positive x axis (exact value)? What is the angle between the vector wnte a ample (Counterlexample on the Same mm as eaCh Wrong formma' Z and the positive I axis (approximate value)? 7 (Finding the wrong formulas will therefore allow you to treat question 2, even if you cannot give the proof of the correct formula.) q = Wall... ‘odrveew \onW‘M. x 41w} aw}. U=<6m65oIébg‘So7‘ .w: sinz+cosz=\/§cos(2z) No.TaK:X=0 0")7L0—9:
I o sinr+cosr = x/Esin(a:+§) yes I puma “94°”,
MM Ge (goth: MA .58“: Ix: $5 Sinx+cos$ : ﬁcos(z+§) ND. Taw‘ : Ll: i614. ‘V'Lj; 0 sin z + cos I sin(2z) Nayakgygo 0H?!)
'w: < ecu>652 “Ms”? L. < 2.54 , 1L+#> Proof: Isa. (VG1‘33: bunk on? i. obiAQLJZ in(¢w I'dCaax)’ _, 1 )b‘ )HE : lb. +Cw=
i»: Mauve“ vw w Ac 0? M A w x , N 2.4L:
‘1wa x outta. \‘w {5  If“ 2) Using the answer to question 1), solve 1
sin$+cosz = ——, with 0515277. 5° (311‘38. ﬂ Keep exact values.  I
Mt»th \) we QAavKhs AdoraL by“ (XiEX: ~2  Self 9: YtLr . W1 Acron Av; 92%;. We. hnokf‘
L‘. 2) A boat whose speed (relative to water) is 6 mi/h maintains a compass heading of 25° _
(i.e. 25" east of north), on a river ﬂowing south with a current of 3 mi/h. In which bmxw 9 ; JI . Oak)“ 0"!» an Q‘Ve'“ '
direction is the boat actually going? (Although you can treat this question independently, 0‘ 6
you can also use the result of the previous question.) .n. ° . . Ir 31211 —— 1+th Ea—Lnr . , Wt CM. um. WWJNV‘V alooVQ \“uapquh/‘Vcﬂoa‘ﬂ hawk/way“ 9 “'J 5 ' ’ 6 ) 6 ’6 )
\‘X’ n n ., Wave, [aqmvux _  “T: 2 M t w M a ’ i * at
d —7 —‘) . x 0" a . ,_ \ 1
Ta Weill/21W 2=VW”&* 3") on F\‘waM4meaV{mX:G”IrEf‘ “swam/Wm ‘m ~ ' d2; aura/a [5 aloovﬂ.
Th: dAV‘CJRm/s v3 onyA‘M ‘
“CAL®% 0% L‘s25 W! ak)‘, X: —2:+21T—FLT :@ 3 OPX21—I:
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 Spring '07
 ROSAY
 Trigonometry

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