082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

3.7 Modelling waves using sint and cost 143
(g) sin(2πt + π) relative to sin2πt
(h) 3cos(5πt −3) relative to 3cos5πt
( )
πt
(i) sin
3 +2 relative to sin πt
3
(j) cos(3π −t)relative to cost
6 Writeeachofthe followingin the form
Asin(3t+θ),θ 0:
(a) 2sin3t +3cos3t
(b) cos3t −2sin3t
(c) sin3t −4cos3t
(d) −cos3t −4sin3t
7 Writeeachofthe followingin the form
Acos(t−θ),θ 0:
(a) 2sint−3cost
(b) 9sint+6cost
(c) 4cost−sint
(d) 3sint
8 Writeeachofthe followingexpressions in the form
(i)Asin(ωt + θ),(ii)Asin(ωt − θ),
(iii)Acos(ωt + θ),(iv)Acos(ωt − θ)where θ 0:
(a) 5sint+4cost
(c) 4sin2t −6cos2t
(b) −2sin3t+2cos3t
(d) −sin5t −3cos5t
Solutions
1 (a) 3,2, 1 π ,0,0 (b) 1
2 ,4, 2 π ,0,0
(c) 1,1, 1 3
,1,1 (d) 4,3,
2π 2π ,0,0
(e) 2,1, 1
1
, −3, −3 (f) 5,0.4,
2π 5π ,0,0
(g) 1,100π,50,0,0 (h) 6,5, 5
2π ,2,0.4
(i)
2 (a)
(d)
2
3 ,0.5, 1
4π ,0,0
2π
7
2π
3
(b) π
(e) π
(j) 4,π,1 , −20, −20
2 π
(c) 2π
(f)
3π
2
3 (a) 15 (b) 20π (c) 0.1
1
(d) 4 (e) (f) −15
5π
2
4
3 sin(πt +k)or 2 cos(πt +k)
3
5 (a) 3, 3 (b) −3, − 3 2
4
(d) −2, −2 (e)
5 , 4 3
(g) π, 1 (h) −3, − 3
2 5π
(j) −3π,−3π
√
6 (a) 13sin(3t +0.9828)
(b) √ 5sin(3t +2.6779)
√
(c) 17sin(3t +4.9574)
(d) √ 17sin(3t +3.3866)
(c) 0.2, 0.4
(f) −0.858,−0.286
(i) 2, 6
π
√
7 (a) 13cos(t −2.5536)
√
(b) 117cos(t −0.9828)
√
(c) 17cos(t −6.0382)
( )
(d) 3cos t − π 2
8 (a) (i) √ 41sin(t +0.675)
(ii) √ 41sin(t −5.608)
(iii) √ 41cos(t +5.387)
(iv) √ 41cos(t −0.896)
( )
√
(b) (i) 8sin 3t + 3π 4
(ii) √ ( )
8sin 3t − 5π 4
( )
(iii) √ 8cos
(
3t + π 4
)
(iv) √ 8cos 3t − 7π 4
√
(c) (i) 52sin(2t +5.300)
(ii) √ 52sin(2t −0.983)
(iii) √ 52cos(2t +3.730)
(iv) √ 52cos(2t −2.554)
√
(d) (i) 10sin(5t +4.391)
(ii) √ 10sin(5t −1.893)
(iii) √ 10cos(5t +2.820)
(iv) √ 10cos(5t −3.463)