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Precalculus 4.1 Angles and Radian Measure
MULTIPLE CHOICE. 01oose the one alternative that best completes the statement or answers the queStion.
The given angle is in standard position. Determine the quadrant in which the angle lies.
1) 53°
A) Quadrant I
B) Quadrant II
q
l) AD) Quadrant Hf
Quadrant IV
2) 124°
2) _ g _
A) Quadrant I
. B) Quadrant II
q
D) Quadrant IV
Quadrant Ill
3) _B_
3) -6()0
A) Quadrant III
3, 0 + (-'
q
. B) Quadrant IV
0
)
::.
-3 oc,
Oassify the angle as acute, right, obtuse, or straight.
4) 26°
A) straight
B) obtuse
5) 115.982°
A) right
n ~ I SO
6)- ,, 6
q
q
B) straight
-:.
D) Quadrant I
Quadrant II
q
B) straight
5)
D
D) obtuse
acute
6)
C
7)
C
D) right
acute
.
. tercepts an arc of length s.
Find the radian measure of the central angle of a crrcle
of radi us r tha t m
7) r = 4 inches, s = 8 inches
1
.
B) - 2 radians
q
2 radians
D)-rad1ans
8) r =2.5 meters, s =4.25 meters
.
A) 1_7 radians
B) 2.2 radians
q
0.4 radians
D) 1.25 radians
A) 2°
D
D) acute
right
30~
'
A) obtuse
4)
2
Gr :. --ts
G -:.-:
_j,_::: ~ ~ / · 7
r
2 -- 6
8)
rM 1· c..i,-\.-
. d egrees to radians. Express answer as a multiple of n.
Convert the angle m
9) 60°
A) n rad.1ans
2
er
Go
?r"-.:: ffi
n
B) ~
3 radians
(3 r :: ~ r ,
I SD
q 5 ra
:: -:;- Tl ::
::,
d.
ians
-11Gv-
::.
9) _B__
n d.1ans
D)-ra
4
I!.
3
10) .Jf__
10) -30°
n d 1ans
.
A)--ra
8
A
n d.1ans
B) - -ra
6
n d.1ans
q--ra
5
Mr.Ashraf S.Maysara
n ct·1ans
D)--ra
7
Convert the angle in radians to degrees.
11)
11)~
J)_
4
A)1 o
D) 45°
B) 45n°
12) -A-
n
12)-4
A) -45°
B) -1°
q
-45n°
Convert the angle in radians to degrees. Round to two decimal places.
13) 3 radians
A) 173.14°
B) -0.11°
q 171.89°
13)
c_
D) 0.05°
;3
= -rr
Draw the angle in standard position.
14) 2n
14) . A c)
:; 1:.20
3
A)
B)
q
D)
2
15) -120°
1s)
A)
B)
q
-A _
D)
Find a positive angle less than 360° or 2n that is coterminal with the given angle.
16) -137°
A) 403°
B) 223°
q 43°
~,i, + (.- \~1)
17) 668°
A) 488°
B) 308°
6£i - 36l)" ==
q334o
TT
17)
13
18)
_Q_
D) 298°
30$J
l8) 17n
8
B) 9n
A)-8
B
16)
D) 137°
8
q 7rc
8
D)~
8
Find the length of the arc on a circle of radius r intercepted by a central angle 0. Round answer to two decimal places.
19) r =7 centimeters, 0 =55°
19)
A) 6.05 centimeters
B) 7.39 centimeters
q 6.72 centimeters
D) 5.38 centimeters
C
6S
;..!!.rz
Jjb
3,
20) r = 11.41 inches, 0 = 60°
A) 12.15 inches
B) 12.05 inches
q
20)
12.25 inches
D) 11.95 inches
D
Solve the problem.
21) The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 40
minutes? If.necessary, round the answer to two decimal
J
. paces.
A) 34.74 inches
B) 33.51 inches
q 36.02 inches
D) 31.77 inches
s -22) A pendulum swings through an angle of 25° each second. If the pendulum is 60 inches long, how
far does its tip move each second? If necessary, round the answer to two decimal places.
A) 24.33 inches
B) 28.61 inches
q 27.47 inches
D) 26.18 inches
--
Gr ...
1'1'
~r -
0
2l) _J3_
22)
_Q_
23)
A
1=S
.-,
1'bD
2.
-~6
TT
Express the angular speed in radians per second.
23) 168 revolutions per second
. A) 336n radians per second
q 168n radians per second
I6~
B) 168 radians per second
D) 336 radians per second
X 21T
4
Precalculus 4.5 Graphs of Sine and Cosil1e Functions
5t
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the que ion.
Detennine the amplitude or period as requested.
1) Amplitude of y = -2 sin x
A) 2n
B) ;
l) _e_
D) -27t
q2
-C
2)
2) Period of y = sin Sx
B) 1
A)S
q 2n
D)2n
5
3) .ll-
3) Period of y = -3 sin x
B) 2rr
A)3.
D)n
q3
3
4) Period of y = 3 sin (8rrx + 4rr)
5) Amplitude of y
1
=3
j2_
5)
-A
1
B)3.
4
A)4
4)
D)4
qsn
cos 4x
D) ..!!_
4
q6n
A)_!_
3
6)
6) Amplitude of y
. 1
=4 sm 3 x
B) 6n
A)4
3
D)~
q5.
4
y ':, A ';,
I' ,' V\,
I Al
fu,·od --
2..1l
g
Mr.Ashraf S.Maysara
A
h the function.
.
7)y = smx
GtaP
3 y
2
-2ll
-ll
7t
21!
-I
X
-2
-3
A)
B)
y
3
y
3
2
2
1
1
-l ir
-ll
lt
-l
-2
-3
-3
q
2ll
D)
3
y
3
l
2
211
-211
y
1
1
-2
-l
-3
-3
\
l
2
8) y
= COS X
3
y
8)
2
-2n
-It
It
-1
2n
l<
-2
-3
A)
B)
y
3
3
y
2
2
X
It
-1
-2
2
X
-2
.3
-3
q
D)
3
y
3
2
-2Jt
·1t
y
:z
It
21t
X
It
·K
-2
-2
.3
.3
3
21t
X
C
Graph the function and y = cos x in the same rectangular system for O:-; x :-; 2n.
9) y = 3 COS X
9) ~
y
4
3
2
n
-1
2n
X
-2
-3
-4
A)
B)
y
y
4
4
3
3
2
2
X
X
-1
-1
-2
-2
-3
-3
--4
-4
D)
q
y
y
4
4
3
3
2
2
X
1
-1
-1
-2
-2
-3
-3
-4
--4
Graph the function.
4
- - -- - - -- --
lO) _B__
10) y = - 3 cos 2x
y
3
-2,r
- 1I
2,r
ll
X
-3
A)
B)
y
y
3
-2,r
X
2,r
q
D)
y
y
3
,r
- 1I
X
X
-3
5
vse a vertical shift to graph the function.
11) y = 1 + sin x
11)
y:,
3 y
2
-21t
-rr
It
2,r
X
-1
-2
-3
A)
B)
3
y
3
y
2
-21t
- It
l[
21t
X
-2
rr
rr
-1
-2
-2
-3
-3
D)
q
3
y
3
2
2
-21t
It
·lt
21[
X
-21[
-l[
-1
-2
-2
-3
-3
6
y
2
X
_c_
--12) y = Sin X - 1
12)
J
1
y
2
-2Jt
-Jt
Jt
-1
2Jt
X
-2
-3
A)
B)
3
y
3
-21t
-Jl
I(
21[
y
X
-2x
-1
-1(
l[
2n
X
-1
-2
-2
-3
-3
q
D)
3
y
3
2
y
2
-21t
l[
.3
I[
-3
7
2ll
X
.c__
l3) y= - 3 sin (3x_+
1
;J-
2
13) A
y
6
4
2
.,r
,r
-2
X
-4
-6
A)
B)
6
y
6
y
4
2
-2
-4
-6
-6
q
D)
6
y
6
y
4
2
X
l[
X
-2
-4
-6
-6
8
I
I
I
Precalculus 4.7 Inverse Trigonometric Functions
,wUL11PLE CHOICE. Otoose the one alternative that best completes the statement or answers the question.
IC
f
find the exact value of the expression.
l) _A_
1) sin- 1 ..[2
2
B) 2n
3
A) 2!.
4
2) cos- 1
A)
D) 3n
4
2!.
q
3
J)_
2)
,fj
2
B) 11n
6
2!.
4
q 7n
4
D) 2!.
6
D
3)
3) tan-11
B) 2n
3
A) Sn
4
q
D) 2!.
4
2!.
3
Use a calculator to find the value of the expression rounded to two decimal places.
4) 1an-l (0.9)
A) 0.84
8) 48.01
q0.73
C
4)
D) 41.99
C
5)
5) sin-1(-¼)
8) 1.74
A) 99.59
1
6) cos- (-
fJ
A)2.53
q -0.17
D)-9.59
A
6)
B) - 0.96
q 144.74
D) -54.74
Find the exact value of the expression, if possible. Do not use a calculator.
7) _ ] _
4
7) sin- 1 [sin [ ; )]
B) 3n
7
A) .l_
3n
8) tan-1 [tan [
4n
A) - -
5
q
7
4n
D)4n
7
8)
4
1t)]
5
TI
B) - -
5
q~
5
Mr.Ashraf S.Maysara
D) 4n
5
1.L_
\I
B) 4n
7t
Q-3
3
D) E_
3
Use a sketch to find the exact value of the expression.
10) cos[sm-1
:J
.
10> A
B)..!._
A)~
5
11) sec[tan- 1
r:
4
D)-5
C-
11) - - -
q
B) _!_
2-✓
3
2
to-,h -l ( lfi
~
S-ec.,
5
fJ
A)2
i'-
3
q--
5
~
)
-
3
D) -✓
3
T1
r:,
-.fi
?
2:---:---
Solve the problem.
= tan-1 (wC/G) gives the phase angle of impedance in the parallel portion of a
distributed constant circuit. Find 02 if w = 15 0 radians per second, C = 0.08 µF per kilometer, and
G = 1.85 µsiemens per kilometer.
U) The equation 02
A) 802°
q
B) 89.1°
0
2
1.40
D) 88.6°
12)
A
Precalculus 4.7 Inverse Trigonometric Functions Part 2
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
.
.
l
. th
.
.
.
.
. .
d . the domain of the
Use a nght tnang e to wnte e expression as an algebraic expression. Assume that x 1s positive an m
given inverse trigonometric function.
1)
1) sin(tan- 1 x)
on,
A
Tl ,
~
A)x~
B)
x2 + 1
x✓x2 + 1
r'I
- I
V T~h
)(
2) - - E 2) cos(tan- 1 x)
A)
x{x2 + 1
B)
x2+ 1
✓x2+1
x2 + 1
I
6
:;:
C)
1t.th- "'X
~x2-1
D) x✓x2 + 1
x2-1
13 (I½~
X
I
f tlh
C..,
0
f)-:.
.P--
V\-)" f
I
V!-t')(
5 l 1C..k-'-X J
-- eos e -
::::
l
I-,. X Z.
~
~
vr.:;x2.
-
VJ -t;. 2
2.
'L
-
Mr.Ashraf S.Maysara
----
~~'4
I -t- X --z.
3) sin(tan- 1
7s)
x✓x2- 5
B)
A) _x.,_
,2 -_ 5-
x✓x2 + 5
C)
x✓x2 + 5
D) _x.,__2_+5-
✓x2 + 5
x2 + 5
!ZJ
X
t,'if,.vk_v33" ~
,Jff..-t;
✓x2+25
4)
4) tan(sec-1 ...:..--)
X
B) ✓x2+5
A)~
x2 + 5
X
C) x ~
x2+ 25
D) 5x
hyf
Q ::::
-t t.l.~
(.
~
- I
✓e C
5~c..
,f;-Qn e
,,. I
-==-
O
S ~c.. (J
fa½-~~
-;:
)(. j
J;:.,f -i _
f
2
a,dJ
2.
4flJ
A
~
,
_erecalculus 4.8 Applications of Trigonometric Functions
MtJL11PLE CHOICE. Otoose the one alternative that best completes the statement or answers the question.
.
.
.
.
o the nearest tenth
Solve the nght triangle shown m the figure. Round lengths to one decimal place and express angles t
of a degree.
B
C
a
A'--- -- - -~c
b
l)_A~
=
=
B) B 500, a= 61.7, c 80.6
D) B = 400, a= 39.7, c = 435
a::::
2) b = 150, c =430
A) A= 1910, B = 70.80, a= 403
q A= 69.60, B = 20.40, a = 403
o..., ::
:: ~
-~
1 50
B) A= 70.80, B = 19.2;
D) A = 69.60, B= 20.4 , a -
v[_ 4-;;)/· - (. l c; o >1.
~ ~ o 7,
!,11
CJ O
-
,$~
~\
c,.o s A-=- ~
g~
C
2)
Go 5
--.
bq · 6
~
;J_ 0
::
ioo
;::.
., '1
f cliff is 13059•. If the base of the cli [f is
1 e the problem.
an le of elevation to the top o a ,
~So v
F
a boat on the lake, the _g . th cliff (to the nearest foot).
D) 408 feet
3) ro88rnf t from the boat, how high is e
q 395 feet
15
ee
B) 405 feet
A) 398 feet
3)
(
r
'X
t
0
I~ cg~
'T(M\, l"; 5<f
:: ~4 5
\.,
1
Mr.Ashraf S.Maysara
4) from a boat on the river below a darn th
darn is 1530 feet above the level of th ' . e a~gle of elevation lo the top of the dam is 14°41 '. If the
nearest foot)?
e river, ow far is the boat from the base of the dam (to the
A) 5809 feet
B) 5819 feet
q 5839 feet
Ill~ 14 °~I
D) 5829 feet
t>
,-
X
1CA.~ l'i
·x
0
4(
I S~o
~
-=-
'SB
3~ .. q
-
,,,...._,,
S g3q
5)
) A radio transmission tower is 240 feet tall. How long should a guy wire be if it is to be attached 14
feet from the top and is to make an angle of 23° with the ground? Give your answer to the nearest
tenth of a foot.
A) 260.7 feet
_c_
IS 3o
15 3~
5
4)
q
B) 245.5 feet
C
D) 614.2 feet
578.4 feet
l'l/i\ ,2.l{O - l"t-:J..26
'
_ 5,1123 =
X
x·
J.2{,
X'
--::::.
5 J 8-
Y
Use the given figure to solve the problem.
6) Find the bearing from O to A.
6)
N
A
B
D
s
A) N 57° E
B) S87° E
@_N33°E
2
D) N 147° E
C
/
Using a calculator, solve the folio .
Wing problems Ro
7) A boat leaves the entrance f h bo · und your answers to the nearest tenth.
miles north and how man °m~l ar r a nd travels 93 miles on a bearing of N 43° E. How many
A) 63.4 miles north an/68 ~ east from the harbor has the boat traveled?
q 93 miles north and 93
es eaSl
B) 68 miles north and 63.4 miles east
mi es east
D) 99.7 miles north and 86.7 miles east
7
An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) and then
released. Write an equation for the distance of the object from its rest position after t seconds.
8)
8) amplitude = 8 an; period = 5 seconds
A) d
d
--
q
B) d = -8 sin l rtl
= -5 cos.!.
rtt
4
5
~
cos
lN
-
.2. Tl
pe,,r,ocl
2
d = -8 cos 5
·t
()...,
- 1S
L.v
J ,..
_.
'D e,oS
2 tl.
~
'j
3
+
TII
D) d = -8 cos
TI
5t
C