Chapter
7
Trigonometric Functions
7.8
1.
Phase Shift; Sinusoidal Curve Fitting
y = 4 s i n ( 2x − π)
Amplitude: A = 4 = 4
2π 2π
Period:
T=
= 2 =π
Phase Shift:
2.
y = 3sin(3x − π)
Amplitude: A = 3 = 3
2π 2π
Period:
T=
= 3
Phase Shift:
3.
= π
2
= π
3
π
y = 2cos 3x + 2 
Amplitude: A = 2 = 2
Period:
Phase Shift :
2π
3
 π
− 
π
 2
=
=−
3
6
T=
2π
=
738
Section 7.8
4.
y = 3cos( 2x + π)
Amplitude: A = 3 = 3
2π 2π
Period:
T=
= 2 =π
= −π = − π
2
2
Phase Shift:
5.
π
y = −3sin  2x + 2 
Amplitude: A = − 3 = 3
Period:
2π
=π
2
 π
− 
π
 2
=
=−
2
4
T=
Phase Shift:
6.
Phase Shift: Sinusoidal Curve Fitting
2π
=
 − π
y = − 2 c o s2x

2
Amplitude: A = − 2 = 2
Period:
T=
Phase Shift:
2π
=
2π
=π
2
π 
 
2 π
=
=
2
4
739
Chapter 7
7.
y = 4sin( πx + 2)
Amplitude: A = 4 = 4
Period:
Phase Shift :
8.
2π
=
=−
2
π
T=
2π
=2
π
y = 2cos(2πx + 4)
Amplitude: A = 2 = 2
Period:
Phase Shift :
9.
Trigonometric Functions
2π
=1
2π
−4
2
=
=−
2π
π
T=
2π
=
y = 3cos(πx − 2)
Amplitude: A = 3 = 3
Period:
Phase Shift :
T=
2π
=
=
2π
=2
π
2
π
740
Section 7.8
10.
Phase Shift: Sinusoidal Curve Fitting
y = 2cos(2πx − 4)
Amplitude: A = 2 = 2
2π 2π
Period:
T=
= 2π = 1
= 4 = 2
2π π
Phase Shift:


π
π
11. y = 3sin−2x + = −3sin 2x − 


2
2
Amplitude: A = − 3 = 3
Period:
T=
Phase Shift:
12.
=
2π
=π
2
π 
 
2 π
=
=
2
4

π
y = 3cos−2x + 

2
Amplitude: A = 3 = 3
Period:
2π
=π
2
 π
− 
 2 π
=
=
−2
4
T=
Phase Shift:
13.
2π
2π
=
2π 2π
= T = π = 2;
1
y = 2sin(2 x − 1) = 2sin 2 x − 2  


A = 2; T = π;
1
= 2;
1
= 2 =2 →
=1
741
Chapter 7
Trigonometric Functions
14.
π
A = 3; T = ;
2
15.
A = 3; T = 3π;
2π 2π
=
= 4;
T  π
 
2 
y = 3sin(4 x − 8) = 3sin [4( x − 2 )]
=
16.
2
3
= 2;
=
4
=2 →
=8
1
= − 3;
1
= −3 →
2π 2π 2
= T = 3π = 3 ;
1 2
2
2
2
2
1
= −3⋅ 3 = −9
y = 3sin  3 x + 9  = 3sin  3  x + 3  


2π 2π
= T = π = 2;
y = 2sin(2 x + 4) = 2sin [2( x + 2 )]
A = 2; T = π;
=
= − 2;
= 2 = −2 →
= −4
π
17. I = 120sin  30π t − 3  , t ≥ 0
2π
1
=
30π 15
Amplitude: A = 120 =120
π 
 
 3 1
Phase Shift : =
=
30π 90
T=
Period:
2π
=
π
18. I = 220sin  60π t − 6  , t ≥ 0
2π
1
=
60π 30
Amplitude: A = 220 = 220
π 
 
 6
1
Phase Shift : =
=
60π 360
T=
Period:
19.
(a)
2π
=
Draw a scatter diagram:
60
0
13
20
(b)
Amplitude: A =
56.0 − 24.2 31.8
= 2 = 15.9
2
742
Section 7.8
Vertical Shift:
(c)
Phase Shift: Sinusoidal Curve Fitting
5 6 . 0 + 2 4 . 2 80.2
=
= 40.1
2
2
2π π
= 12 = 6
Phase shift (usey = 24.2, x = 1):
π

24.2 =15.9sin ⋅1−  + 40.1
6

π

π

π π
−15.9 = 15.9sin
 −  → −1 = sin −  → − = −
6

6

2 6
2π
=
3
π
2π
Thus, y = 15.9sin 6 x − 3  + 40.1
(e)
60
0
20.
60
13
20
0
(d)
y = 15.62sin(0.517
x − 2 . 0 9 6 )+ 40.377
(a)
Draw a scatter diagram:
13
20
85
0
(b)
30
13
80.0 − 34.6 45.4
Amplitude: A =
= 2 = 22.7
2
Vertical Shift: 8 0 . 0 + 3 4 . 6= 114.6 = 57.3
2
2
2π
π
=
=
12 6
Phase shift (usey = 34.6, x = 1):
π

34.6 = 22.7sin  ⋅1−  + 57.3
6

π

π

π π
−22.7 = 22.7sin −  → −1 = sin −  → − = −
6

6

2 6
2π
=
3
π
2π
Thus, y = 22.7sin  6 x − 3  + 57.3
743
Chapter 7
Trigonometric Functions
(c)
(e)
85
85
0
21.
13
30
0
(d)
y = 22.6128sin(0.5032x − 2.0384)+ 57.1686
(a)
Draw a scatter diagram:
30
13
80
0
13
20
(b)
(c)
75.4 − 25.5 49.9
Amplitude: A =
= 2 = 24.95
2
5 100.9 = 50.45
Vertical Shift: 7 5 . 4 + 2 5 . =
2
2
2π
π
=
=
12 6
Phase shift (usey = 25.5, x = 1):
π

25.5 = 24.95sin  ⋅1−  + 50.45
6

π

π

π π
−24.95 = 24.95sin −  → −1 = sin −  → − = −
6

6

2 6
2π
=
3
π
2π
Thus, y = 24.95sin  6 x − 3  + 50.45
(e)
80
80
13
0
20
(d)
13
0
20
y = 25.693sin(0.476
x − 1.814) + 49.854
744
Section 7.8
22.
(a)
Phase Shift: Sinusoidal Curve Fitting
Draw a scatter diagram:
80
0
(b)
(c)
13
30
77.0 − 31.8 45.2
Amplitude: A =
= 2 = 22.6
2
Vertical Shift: 7 7 . 0 + 3 1 . 8= 108.8 = 54.4
2
2
2π
π
=
=
12 6
Phase shift (usey = 31.8, x = 1):
π

31.8 = 22.6sin  ⋅1−  + 54.4
6

π

π

π π
−22.6 = 22.6sin −  → −1 = sin −  → − = −
6

6

2 6
2π
=
3
π
2π
Thus, y = 22.6sin  6 x − 3  + 54.4
(e)
80
0
80
13
30
23.
0
13
30
(d)
y = 22.4587sin(0.5058
x − 2.0602)+ 54.3482
(a)
3.6333 + 12.5 = 16.1333 hours which is at 4:08 p.m.
8.2 − (− 0.6) 8.8
Amplitude: A =
= 2 = 4.4
2
8.2 + (− 0.6) 7.6
Vertical Shift:
=
= 3.8
2
2
2π
π
= 12.5 = 6.25
Phase shift (usey = − 0.6, x = 10.1333):
 π

−0.6 = 4.4sin
⋅10.1333−  + 3.8
 6.25

 π

10.1333π

−4.4 = 4.4sin
⋅10.1333−  → −1 = sin
− 
 6.25

 6.25

(b)
745
Chapter 7
Trigonometric Functions
π 10.1333π
=
− → = 6.6643
2
6.25
π
Thus, y = 4 . 4 s i n 6.25 x − 6.6643 + 3.8
−
(c)
9
0
12
–1
(d)
24.
(a)
(b)
π
y = 4 . 4 s i n 6.25 (16.1333) − 6.6643  + 3.8 = 8.2 feet
8.1833 + 12.5 = 20.6833 hours which is at 8:41 p.m.
13.2 − 2.2 11
Amplitude: A =
= 2 = 5.5
2
13.2 + 2.2 15.4
Vertical Shift:
=
= 7.7
2
2
2π
π
= 12.5 = 6.25
Phase shift (usey = 2.2, x = 14.2333):
 π

2.2 = 5.5sin 
⋅14.2333−  + 7.7
 6.25

 π

14.2333π

−5.5 = 5.5sin
⋅14.2333−  → −1 = sin
− 
 6.25

 6.25

π 14.2333π
− =
− → = 8.7252
2
6.25
π
Thus, y = 5.5sin  6.25 x − 8.7252  + 7.7
(c)
15
0
(d)
0
15
π
y = 5.5sin  6.25 (20.6833)− 8.7252  + 7.7 = 13.2 feet
746
Section 7.8
25.
(a)
Phase Shift: Sinusoidal Curve Fitting
12.75 − 10.583 2.167
Amplitude: A =
= 2 = 1.0835
2
12.75 + 10.583 23.333
Vertical Shift:
=
= 11.6665
2
2
2π
= 365
Phase shift (usey = 10.583, x = 356):
 2π

10.583 =1.0835sin 
⋅ 356 −  +11.6665
 365

 2π

 712π

−1.0835 = 1.0835sin
⋅ 356 −  → −1= sin
− 
 365

 365

π 712π
− =
− → = 7.6991
2
365
2π
Thus, y = 1.0835sin  365 x − 7.6991 + 11.6665
(b)
13
0
10
(c)
26.
(a)
400
2π
y = 1.0835sin  365 (92)− 7.6991 + 11.6665 = 11.85 hours
13.65 − 9.067 4.583
Amplitude: A =
= 2 = 2.2915
2
13.65 + 9.067 22.717
Vertical Shift:
=
= 11.3585
2
2
2π
= 365
Phase shift (usey = 9.067, x = 356):
 2π

9.067 = 2.2915sin 
⋅ 356 −  +11.3585
 365

 2π

 712π

−2.2915 = 2.2915sin
⋅ 356 −  → −1= sin
− 
 365

 365

π 712π
− =
− → = 7.6991
2
365
2π
Thus, y = 2.2915sin  365 x − 7.6991 + 11.3585
747
Chapter 7
Trigonometric Functions
(b)
15
0
(c)
27.
(a)
400
8
2π
y = 2.2915sin  365 (92)− 7.6991 + 11.3585 = 11.7 hours
16.233 − 5.45 10.783
Amplitude: A =
=
2
2 = 5.3915
16.233 + 5.45 21.683
Vertical Shift:
=
= 10.8415
2
2
2π
= 365
Phase shift (usey = 5.45, x = 356):
 2π

5.45 = 5.3915sin 
⋅ 356 −  +10.8415
 365

 2π

 712π

−5.3915 = 5.3915sin
⋅ 356 −  → −1= sin
− 
 365

 365

π 712π
− =
− → = 7.6991
2
365
2π
Thus, y = 5.3915sin  365 x − 7.6991 + 10.8415
(b)
17
0
(c)
28.
(a)
400
5
2π
y = 5.3915sin  365 (92)− 7.6991 + 10.8415 = 11.74 hours
12.767 − 10.783 1.984
Amplitude: A =
= 2 = 0.992
2
12.767 + 10.783 23.55
Vertical Shift:
=
= 11.775
2
2
2π
= 365
Phase shift (usey = 10.783, x = 356):
 2π

10.783 = 0.992sin 
⋅ 356 −  +11.775
 365

 2π

−0.992 = 0.992sin
⋅ 356 − 
 365

748
Section 7.8
Phase Shift: Sinusoidal Curve Fitting
 712π 
π 712π
−1= sin
−  →− =
− → = 7.6991
 365

2 365
2π
Thus, y = 0.992sin 365 x − 7.6991 + 11.775
(b)
13
0
10
(c)
400
2π
y = 0.992sin 365 (92)− 7.6991 + 11.775 = 11.9 hours
749