MAC 1147 Exam #5b
Name:
ID#
&swer KtJ
St(l\m.1r
cloll
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Instructions: Do all scratch work on the test itself. Make sure your final answers
are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON T HIS EXAM IN ORDER TO RECEIVE FULL CREDIT!l!
No.
Score
1
2
/12
3
/ 10
/ 10
4
5
6
7
8
/4
/4
/ 10
/8
/10
/8
9
10
Bonus
/24
/ 10
I Tot al I
/ 100 I
(1) Find the exact value of each expression. (4 points each)
(i) cos- 1 (cos
5 11")
3
(b)
(a) -~
(d)
5 11" 3
2;
(e) None of the
above
1r
- 3
(ii)
;;;eJS"(~
~)
(a) ~
(b) ~
(d) Undefined (e) None of the
above
e
~
(iii) sec [sin- 1 (-~)]
~
() 7
\,J'V45
45
(b) \145
7
(c) -
vk
(d) -
Vf5
(e) None of the
above
+
~~
'tfb';jW ~~
J ~ 1'[.
(2) Write the trigonometric expression as an algebraic expression in u. (4 points)
(a) Jl-u 2
u
(d) v'U:-l
(3) Establish the identity (10 pomts)
.
cos x
. x
1 + sm
+ 1 + sin x
cos x
=
2 sec x
COs x
M."\1iP~ f\~aJtr tr
Co5L-)C
+
\~,,~
tos
c,s X (1+SIt\X)
'X
~~ot
faUi~ ~ eo.s ~
V,~
l-~(\2...
~,
)(
to~ '( (I+s~ 'X.)
O>s'><~
(..Os )c
~j, Qi{..u sJv"'~ poSS\lJtj
)u ~)(Q~ 1\:-S-~ -Gv
-rSSi~i\~~
-+
~\-!,\..X)
(rl­
(J,S )(
fcttAv
-\­
\h1t'. k Cos~
CW
S~
(4) Find the exact value of each expression. (5 points each)
(i) tan 255
0
:::
~
=:: ~
CmO +30')
'J:J5
0
+- ~ ~o
~~o _ ~'1)'
~-----
-
\ - ~ 2~S" . .,.,30'0
1-r-~3lXf · ~~)
----,
- (l+ 1) C"J -&- 1
- (1- (I)(~))[~)
I"\' (-Ii') (\)
)+-\fJ
s - {3
~
1Uu ~w(f) I~\--
~ -+W.
(ii) sin 5811"
::: Ua.,)
_ +
- -
\\­
(-!i))c1-)_
~(;.l.)
- ['3 - 1 .
-
-
r
S((rr.L
(- {3
J
cJ.;t.Ifre..t
~
~ ~
(r ~ ~Llt ~
~ CA catWWtw-')
(5) Find the exact value of cos(a
+ (3)
151?7fwtr
- <
tan a = - -
~ ::
(b)
-~
CoS r:J.:;'
--
_ L~
--
-
fs~'o
~S'
< 7fO sm (3 = - - -
~ :;
\
+- foO
~S"
~J 1t
-
41(""37f
°
O!
8' ~ '
,j~
I~
under the given conditionso (4 points)
...L
11­
< 2(3 < 7f
F-~~:~~e
5'
1f
2
~vJ 1f
of the
(6) Establish the identity. (10 points)
cos (a + (3) cos a cos ,8 = 1 - tan a tan 13 _Co5 tI-
Ciȴ -
~i"'~ Si~~
U>S~ ~;)f
_
1­
~ W'\ at s",
t'
CGs rA (.c,~
f
(7) Find the exact value of each expression. (4 points)
(i) tan (tan- 1 ~
(a) ;;....
+ sin- 1 1.)
' ",~bF~i,2vro
~
(c)
2f'
(d)
1~~~1 1(.;\ None of the
~ (~~) -t- -f4.,(....~ ~)
o
above
~ _ (~+ t)(r~ _ qH\r3_
\ _ ~(¥~).~(~.d) - - (I - -H~)){tz\- !'l- 3B
~
J.~
(ii) cos [2
(a)
~in - :-(- £)]
-g
~",J
@i~~
n:
(c)
21:
10
(--E~
\- ~ [,&il\ (~\,,-{ ~))T
\ - d.'OI"'- (<Jlli'
\-at- ~~ (-;}-(,:~i \
_ ~: 110"- -.£L _ lt9 Ib~
rl9'1
fro" - ~
(d) 1Q
13
(e) None of the
above
(8) Establish the identity. (10 points) cos (3x) =
3
COS X -
3 sin2 x cos X CD5 (~ Cos X - 'S\r"'I ('2.~ ~~X~----.;-.!
(\
i.V",'l-X- 'b~II"XP>S lc
Q§," y. - 5'''1.)(
to~
-
~S"'1< (.oS)r. • SI""
'DMk fl_J. ~J" s
.-
7r
UIS-\"Y1'wk
l(
-
d. £oin2.'1(. Co;)c
2Tr
/h.
blr'7
R
(9)
(i) Express the product as a sum containing only sines or cosines. (4 points)
cos (30) cos (50)
(a) ~ [cos (80)
+ cos (20)J
(c) cos2 (150 2 )
(b) Hcos (80) - sin (20)]
@ ~ [cos (20) + cos (80)J
-5 [ Co~ (~-5e) + u..(oe~S(})l
~
[UlS (- ae) ~ CA5 ( ~&11
--------
~~U1~
(ii) Express the sum or difference as a product of sines and/or cosines. (4 points)
sin (80) - sin (20)
(a) 2 sin (60)
®
2 sin (30) cos (50)
(b) 2cos(30)cos(5e)
(d) 2sin(50)cos(30)
(10) (i) Solve the equation on the interval 0 :s; () < 21f. (6 points - 3 points for the
answer and 3 points for t he steps)
4sin 2 e=3
(ii) Solve the equation on the interval 0 :::;
answer and 3 points for the steps)
2 sin (3e)
e < 21f.
=
(6 points - 3 points for the
v'3
n~'. ~I 1~ ,,~r. '+~= ~/~}~= ~ IA _
.\-
(V
~~ t
2ntT
----1>
~
2 ,. t]I.
l YE - /3 Jr 111: + 'i rr =; T '3 -qJ t; -y
'1f
(iii) Solve the equation on the interval 0
answer and 3 points for the steps)
cos2 X
(b) ~
(05~')< - Co:;
(c)
'--v----"
2tos
UJS2..x -
x < 27r. (6 points - 3 points for the
=0
cos (2x)
11"311"
(d) 0
2'2
(O-;,.\--==-I
("l)c) :::-0
2
-
:s:
(e) None of the
above
CO} X ::: ..... ,
X. :: (o5~( (I \
X-::
X -=
X--=--lT
(os-(-I)
JC -\
0
2~x ""'(:::0
(DAY! ~o
t- Co X
tu'x
c..os"LX, -::. 1 ~ Cb.) X ~
±I
(iv) Solve the equation. Give a general formula for all the solutions. (6 points ­
3 points for the answer and 3 points for the steps)
.
sm(2e) =
~e = ~ + n7r, e =
(c)
+ n7r
(b)
3: + 2n7r
(d)
3811"
e = ~ + 2n7r, e =
(e) None of the above
2V2
e = 7!.8 + 2n7T ' e = 37r8 + 2n7r
e = 7!.4 + n7r , e = 311"4 + n7r
Bonus. Write the trigonometric expression as an algebraic expression containing 'U and v .
(10 points)
cos (tan -1 u - cos - 1 v)
L,...."...--
fI..
~
~
lA
(\f"
\J..7- +\"L = C7.­
C.oS d.::-
~\A~ -\-\
Q1.1"- -/lJ"­
=-\
1-
tt ~\ ~ (,
'J\V\
c~~
a1. -r-rv 'L = \"2.
d.. -=
'-" ~\.t -\-1
_
t...
(A.... ::. \ -Nl.
G-:..
[1-1\);­
J!-t\i'L
~Ii\~:: \
-=-w