Trigonometricidentities.
You shouldalsobe farniliar with the following trigonometricidentities:
sin(-l; = -.;tt 7
cos(-l) = cosI
sin(l + B) : sin(l)cos(3) + cos(l)sin(B)
cos(l + B) : cos(l)cos(B)- sin(l)sin(B)
Radian measure.
You areprobablyfamiliar with the degreeasa measureof angle.A radianis anothermeasureof
angle.The anglesubtendedby an arc, in radians,is the arc lengthdivided by the radiusofthe
circle. Sincethe circumferenceof a circle with radiusr is given by 2r r,360 o is equivalentto
2m/r:2nrad. You shouldalsoknow that 180ois equivalentto rrad,90 o is equivalentto z/2
rad,and45ois equivalentto zd4rad.I rad is equivalentto 180/r : 57.30".
.Sinceaa anglein radiansis onelength (the arc length)divided by another(the radius),the radian
is actuallyunitless.However,insteadof writing simply a number,without a unit, for an angle,
tle label "rad" is often appendedto remind you that the angleis given in radians.
Questionsand ExampleProblemsfrom Chapter 3
''
Question1
ln the figure below, are(a) the x componentand (b) the y componentof vector 7 positive or
negative?Are (c) thex componentand(d) they component
of thevectorcombination7 - .F
positive or negative?
7\l
r,\i-Kffiva
(q/
,
/l
\b)
\
L- ^
.Y\tw^ ^)v
*r\t-
G ) n^t4il',""
/,\
- bJ
,A.B
|
ll ) nuzU'mw
Question 2
6* ths mapltude of the diflerencebetweentwo vectorseverbe greaterthan (a) the magnitude
of oneof the vectors,(b) the magnitudesof both vectors,and (c) the magnitudeof their sum?
(o)
n)94
<J
(u) ^u)
(c l
rw
;-
OY
z
t/\
-+
t.\-
-
(5.o. ) / * (5.Dr)i
1-5.o^ ); - ( 5.Dn)J
\
,/!
ProblemI
What are(a) the x componentand @)the y componentof a vector d in the xy planeif its
direction is250o counterclockwisefrom the positive directionof the x axis andits magnitudeis
7.3m?
ex : *(%:o) s,naoo=F^E;l n
c6z'1o" '= -g
O* = -(""3^)
Ay
0,y
a
d=Gt"5-) i* G6s,):
Problem2
Forthevectors6 = Q.0m)i + @.0@j nd E = 6.0 m)i+ (-2.0m)i, gle d + 6 in (a)unitvectornotation,andas (b) a,qaglrtude and(c) an angle(relativeto r";, Now give 6 -ap
(d) unit-vectornotation,andas (e) tragnitude'and(f) an angle.
q)
( r,o'.') ? +{t.o ,)3
G,*t'=
'[-;t'{r*f -Ery
b)
t)g
J) d-= b - q
t--
=
f(s.o"; i+ (-J.oo)l] - [t:.".)i*(r.o"')3] -
=f;"".r+;;'- =f6El
u[ff.J,"
e)J
t) f - ionl'tt*) ----Io =-?e7
Problem3
,;
, .^^
.a
.-^
.1
(a) In nnit-vectornotation,whatis the sumof d = (4.0lz)i + e.Ofi j and
b = (-13.0n)i+ (7.0m)j ?
What are(b) the magnitudeand(c) directionof d + 6-(relativeto r")?
:+
C'= O'*6
(b) e
--
/n
--4
Cx =4**b*
\
c, =oy,b,
/1
t' ^
, ,^
L-y = lout?r + \-It"L
= _g"orn
,-)
C, = 3.aro +,l.om
====
=
|}}.
--)
1,^l
\ur*Ly'
,[G.*)\;')".---
6 = (-9-oc,r)
e* 1to"o-) t
+
O=lJ.5n
(c )
(g*)
O :,r*,'='(* ) =iq,,r-,
'-(
= g" .?
)t4d"q XtuA"^'.t
r---'----'---'-'tr...o
I
ar
lV=tsa
--+_.......*
/
I
l"
"lA
)t0"
Find tlte sup of the following four vectorsin (a) unit-vectornotation,and as(b) a magnitudeand
(c) an anglerelative to +x.
dY
.P: 10.0m, at 25.0"counterclockwise
from +x
from +y
Q: 12.0m, at 10.0"counterclockwise
R : 8.00m, at 20.0oclockwisefrom -y
S : 9.00 m, at 40.00counterclockwiseftom -y
J
,\
n-+
,x9.4-
-
t\
--'r
--
--\
-
P * O* A'* S'
4
r-\'
t--. -
/,n
^
\
.s-
( lu,ef\/ rc44e5.6
t-
=
u.4
t
t^
^
\
-
L +( lo.oro) Ay\45.o"
j1
(9,,orn....,)ln
lq,a:-) S ,,
-__=_.
.'.,
\
-(ta.o*);ID.ort+(n.o
-=
(-)"ot-)l
Y./
:
"
(rr.sr.,)1
- (v.oo-)1m15.o'i(8,oo"r) cae)o.o'j
E=
= (-x.t,l-) a _ (r"sa -)
_i
J
( n.oo.)-zl-."q0.o" t - ( ?.oo.) c c o ' l b . o "
J
J'-
r\
A -
D)
/t-
c)
a-
JIJ'
(s.tz-)t-1a"rr.)j
lD"A r"l
o , 9.as'
Problem5
Vector d hasa magnitudeof 5.0 m andis directedeast.Vector 6- hasa magnitudeof 4.0 m and
is directed35owest of north. What are(a) the magnitudeand (b) the directionof d + 6 ? What
arethe (c) magnitudeand(d) direction of n - a I 61Draw a vector diagramfor each
combination.
._5
C'= (E"o".r)i+ (o*,)J
= -4,o^,orr^n35"
t + '1.o. coc35j
f,
= (-),i*r)i*{:">.")i
+
f --;1s".-) f+ to-) j] + [1-r.a",)1"
1=.:-)5I
o.J-,1'(ffi)
la-"i)
b- o- = f(-r..", ) l*1r-:-)3] - [qs""^)l*fo")3] = (-r"3-r)in(:":.)J
(^tl,,,u^)--tr"
probremo 0=fr.-
=
a /s6"
lA=tsu"|
Inthe sum A+B=C,vector I hasamagnitude
of 12.0m andis angled40.0'
from the+x direction,andvector C hasa magnitudeof 15.0m andis angled
counterclockwise
20.0ocounterclockwisefrom the -x direction.What are(a) the magnitudeand(b) the angle
(relativeto +x) of B .
A = ( lt -or.,)rns1o.o"/ * 1ta.a.) s ,.,! o.D"j
= (q.rc^)?*(y"tt".)j
_---
- (ts.o-) c,otJo"oo - (rc-a-) ta
i
t_
:
*'""
J
(- tt "t^) t - (s"r:.) j
R = a -n- = L Gtr.t^)t -(s" 13",)j
] - tCq.rl.)l+(r.2.)j]
---f
-\
-5r
'l
(-a:.:-)/-(ra.sr)j
R=
v
O =,hr-' (* )
-
(-a:.:-)^*(-r:,r.)n
/
')r''\
+
- -r
-))'t"r \
^
(/ --::-:_-)=')t.9"
\
)-J-t m
'/
B =ac.o.
-------D'3
\
1/,2d44'nl
A : 1 n Qe\v
"
\_.,)
I
\_.,