Precalculus Exam 6 Review Sheet SOLUTIONS
1.
The accompanying diagram shows the path of a cart traveling on a
circular track of radius 2.40 meters. The cart starts at point A and stops
at point B, moving in a counterclockwise direction. What is the length of
minor arc AB, over which the cart traveled?
Express your answer in simplest form in terms of !.
Convert 165º to radians:
"
165" 11"
165º !
=
=
180º 180
12
2.40 =
2.
24 12
=
10 5
s =! "r
$ 11# ' $ 12 '
s=&
"
% 12 )( &% 5 )(
s=
11# 12
"
12 5
s=
11#
meters
5
The accompanying diagram shows two cables of equal length supporting a pole. Both cables are 14
meters long, and they are anchored to points in the ground that are 14 meters apart.
What is the exact height of the pole, in meters?
(1) 7
(3) 7 3
(2) 7 2
30º
Since the triangle is equilateral, each interior angle has a
measure of 60º. Therefore, the pole (the altitude) divides the
triangle into two 30º–60º–90º triangles, one of which is shown
at left. The base of this triangle is 7 since the altitude in an
equilateral triangle bisects the base. The ratio of the sides in
such a triangle is 1 : 3 : 2 . Thus, the height of the pole is
7 3.
14
7 3
60º
7
3.
(4) 14
An art student wants to make a string collage by connecting six equally spaced points on the
circumference of a circle to its center with string. What would be the radian measure of the angle
between two adjacent pieces of string, in simplest form?
A full rotation is 360º or 2! radians.
Each central angle is one-sixth of a full
rotation. Thus, its radian measure is
!
1
2"
! ( 2" ) =
6
6
=
"
3
Or, we could work in degrees and then
convert to radians:
1
"
! ( 360º ) = 60º =
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6
3
4.
As the pendulum of a clock swings through an angle of 70º, the tip of the pendulum travels along
an arc whose length is 2! inches. Find the length of the pendulum.
Convert 70º to radians:
70º "
70"
7"
!
=
=
radians
1 180º
180
18
s =! "r
7#
2# =
"r
18
$ 7# '
18 " ( 2# ) = 18 " &
"r
% 18 )(
36# = 7# " r
r=
5.
r
36
inches
7
70°
r
2!
If sin A > 0 and (sin A)(cos A) < 0 , in which quadrant does !A terminate?
(1) I
(2) II
(3) III
(4) IV
Since sin A > 0 , we know that !A terminates in either Quadrant I or Quadrant II.
Since (sin A)(cos A) < 0 , cos A must be negative. Since cos A is negative in Quadrant II,
!A must terminate in Quadrant II.
6.
If cos 3xº = sin 60º , then x may equal
(1) 10
(3) 30
(2) 20
(4) 40
Since cofunctions of complementary angles are equal (and cosine and sine are cofunctions),
3x + 60 = 90
3x = 30
x = 10
7.
Through how many radians does the minute hand of a clock turn in 24 minutes?
!
5
2!
(2)
5
(1)
3!
5
4!
(4)
5
(3)
Working in degrees:
2
! ( 360º ) = 144º
5
Then, we convert to radians:
144º !
"
144"
4"
=
=
180º 180
5
Since a complete rotation of the minute hand takes 60
minutes, it turns
24
2
or
of a complete rotation
60
5
in 24 minutes. Therefore, the minute hand turns
2
of 2! radians in 24 minutes:
5
Or, working in radians directly:
or
2
4"
! ( 2" ) =
5
5
2
of 360º
5
8.
a. Convert to radians and express in simplest form: 315º
315º "
315"
7"
!
=
=
1 180º
180
4
b. Convert to degrees and express in simplest form:
11!
9
20
11! 180º
11 ! 180 º
"
=
"
= 220º
9
!
9
!
1
9.
Express as a function of a positive acute angle:
a.
b.
10.
11.
sec 935º = sec ( 935º !720º ) = sec 215º = ! sec 35º
18
#
&
7!
7
!
180
º(
%
c.
sin
= sin
"
= sin126º = sin 54º
% 10
10
! (
$ 1
'
Express as a function of a positive acute angle less than 45º:
a.
sin 245º = ! sin 65º = ! cos 25º
b.
sec 98º = ! sec 82º = ! csc 8º
c.
cos 355º = cos 5º
Find the exact value:
a.
sin 315º = ! sin 45º = !
b.
csc150º = csc 30º =
2
2
1
1
=1 = 2
sin 30º
2
cos180º !1
=
undefined
sin180º
0
Express as a single fraction in simplest radical form:
11!
# 5! &
11!
7!
# 5! &
= sin
" cot % " (
sin
" cot % " ( + cos
$ 3'
6
$ 3'
6
4
c.
12.
cot ( !459º ) = cot ( !459º +720º ) = cot 261º = cot 81º
cot180º =
= sin 330º
" cot ( "300º )
7!
4
+ cos 315º
= " sin 30º
" cot 60º
+ cos 45º
1
2
1 # 3&
= " % (
2 $ 3'
"
3
3
3 # 2&
"
% (
3 $ 2'
+
2 3
6
+
= "
=
"3
6
"
+ cos
2
2
2 # 3&
+
% (
2 $ 3'
3 2
6
!3 ! 2 3 + 3 2
6