Download Chapter 5 Test Review

Name
February 2015
Honors Math 3
Chapters 5-Review page 1
Chapter 5 Test Review
Topic outline
The test will include all sections of chapter 5 that we covered (except 5.01 – 5.04, 5.07, 5.08).
Chapter 5: Analyzing Trigonometric Functions
The cosine and sine functions
o Understand the relationship between degree and radian measure
o Calculate sine and cosine using radian measure
o Relate the motion of an object around a circle to the graphs of sine and cosine
o Graph sine and cosine using radian measure
o Solving cosine and sine equations
o Make sense of sinusoidal functions in the context of previous experience
o Understand the geometry of sinusoidal functions
o Model with sinusoidal functions
**Additional review problems on page 444-445 in our textbook!
1. How many radians correspond to an angle with degree measure
?
2. Complete with > , < or = without using a calculator.
a. cos
31p
5p
____ cos
18
18
b. sin
31p
5p
____ sin
18
18
3. Solve each of the equations for the general solution(s) and the solution(s) in the interval
[0,2p ).
a. 2cos x =1
b.
c. sin2 x - 2sin x - 3 = 0
d.
e. sin(cos x) =1
f.
sin3x = sin x
3
2
cos2x + 5cos x = 2
cos2x =
4. If 3cos x + 4 = 3, find all possible values of sinx.
5.
Find a sinusoidal function formula to match the each graph below.
1
Name
February 2015
a.
Honors Math 3
Chapters 5-Review page 2
b.
c.
6. The graph of a sinusoidal function, f (x), is given below. Find 3 function formulas, at least
one using sine and at least one using cosine (remember that x is measured in radians)
2
Name
February 2015
Honors Math 3
Chapters 5-Review page 3
7. George of the Jungle is swinging back and forth on his vine. As he swings, he goes back and
forth across the river bank, going alternately over land and water. Ursula decides to model
mathematically his motion and starts her stopwatch. Let t be the number of seconds the
stopwatch reads and let y be the number of meters that George of the Jungle is away from the
river bank Assume that y varies sinusoidally with t, and that y is positive when George of
the Jungle is over water and negative when he is over land.
Ursula finds that when t = 2, George of the Jungle is at one end of his swing, where
y = -23. She finds that when t = 5 he reaches the other end of his swing and y = 17.
a) Sketch the graph of this sinusoid.
b)Write the equation for your sinusoid above.
c)Predict the number of meters that George of the
Jungle is from the river bank (i.e. y) for
i. t = 2.8
ii.. t = 6.3
iii. t = 15
d)Where was George of the Jungle when Ursula
started her stop watch?
e)Find the least positive value of t for which George
of the Jungle is directly over the river bank (i.e. y = 0).
8. Take it Further You seek a treasure that is buried in the side of a mountain. The mountain
range has a sinusoidal cross section. The valley to the left is filled with water to a depth of 50 m,
and the top of the range is 150 m above the water level. You set up an x-axis at water level and a
y-axis 200 m to the right of the deepest part of the water. The top of the mountain is at x =
400m.
a)Write the equation expressing y in terms of x for points on the surface of the mountain.
b)Show by calculation that this sinusoid contains the origin (0, 0)
c)The treasure is located within the mountain at the point (x, y) = (130, 40).(Note this point is not
on the graph). Which would be a shorter way to dig to the treasure, a horizontal tunnel or a
vertical tunnel? Justify your answer.
3
Name
February 2015
Honors Math 3
Chapters 5-Review page 4
9. A circular saw has a speed of 15,000 radians per minute. How many revolutions per minute
does the saw make? How long will it take the saw to make 6000 revolutions?
Answers
1.
8p
9
2a. =
3a. cos x =
1
2
x=
p
3
+ 2pn x =
(
b. <
5p
p
5p
+ 2pn; x = x =
3
3
3
)
b. sin3x = (sin x) 4 cos2 x -1 therefore this equation becomes
(sin x)(4 cos2 x -1) = sin x
2(sin x)(2cos2 x -1) = 0
sin x = 0 and cos x = ±
2
2
x = pn; x =
p
4
+ pn,
3p
+ pn for any integer n
4
p 3p 5p 7p
x = 0, , , p, ,
4 4
4 4
c.
(sin x - 3)(sin x +1) = 0
sin x = -1
3p
for any integer n ; x =
3p
x=
+ 2p n
2
2
p
11p
+ 2pn
d. 2x = + 2pn,
6
6
p
11p
x = + pn,
+ pn for any integer n
12
12
p 11p 13p 23p
x= ,
,
,
12 12 12 12
e. cos x =
f.
p
2
+ 2p n for any integer n; no solution
2cos2 x -1+ 5cos x - 2 = 2cos2 x + 5cos x - 3 = 0
( 2 cos x -1) ( cos x + 3) = 0
cos x =
x=
p
3
1
2
cos x = -3
+ 2p n,
4. sin x = ±
5p
p 5p
+ 2p n for any integer n ; x = ,
3
3 3
2 2
3
4
Name
February 2015
Honors Math 3
Chapters 5-Review page 5
5. a.
f (x) = 3sin 4x - 2 or f (x) = 3cos(4(x - p8 )) - 2
b.
f (x) = 2sin( x + p4 ) + 3 or f (x) = 2cos( x - p4 ) + 3
c.
f (x) = 4 sin 2( x - p2 ) +1 or f (x) = 4 cos 2( x + p4 ) +1 or f (x) = -4 sin2x +1
(
)
(
)
æp æ
æp ö
æp
ö
5 öö
6. answers vary: f (x) = -5cosç x÷ - 2; f (x) = 5cosç ( x ± 5)÷ - 2; f (x) = 5sinç ç x - ÷÷ - 2
2 øø
è5 ø
è5
ø
è5è
7. a.
æ p æ 7 öö
æp
ö
æp
ö
b. answers vary: y = 20cosç (t - 5)÷ - 3; y = -20cosç (t - 2)÷ - 3; y = 20sinç ç t - ÷÷ - 3
è3
ø
è3
ø
è 3 è 2 øø
c. At t = 2.8, y  -16.38 meters; at t = 6.3, y  1.16 meters; at t = 15, y = -13 meters.
æ -5p ö
d. At t = 0, y = 20cosç
÷ - 3 = 7. So when Ursula started her stopwatch, George is 7
è 3 ø
meters away from the river bank, over the water.
e. 0.36 seconds
8.
æ p
ö
æ p
ö
a. y = 100cosç
( x - 400)÷ + 50 or y = 100sinç ( x -100)÷ + 50
è 600
ø
è 600
ø
b.
æ p
ö
æ 2p ö
æ -1ö
f (0) =100cosç
(0 - 400)÷ + 50 = 100cosç - ÷ + 50 = 100× ç ÷ + 50 = 0
è 600
ø
è 3ø
è2ø
c. Vertical Tunnel:
let x = 130
æ p
ö
f (130) = 100cosç
(130 - 400)÷ + 50 = 65.64
è 600
ø
Horizontal Tunnel:
Tunnel = 65.64 - 40 = 25.6
let y = 40
æ p
ö
40 = 100cosç
( x - 400)÷ + 50
è 600
ø
x = 719.1+1200n or x = 80.7 +1200n
The first x value is about 80.7 meters therefore the tunnel = 120 – 80.7 =49.3 m
Vertical Tunnel is about 23.7 meters shorter.
æ
rad öæ 1 rev ö 15000
rev
9. ç15000
» 2387.3
֍
÷=
è
min øè 2p rad ø
2p
min
2387.3
rev 6000 rev
; 2.51 minutes
=
min
x min
5