Answer
a)
$V=31.25\cos(25t)\ V$
b)
$V=-125\cos(50t+\pi/2)=125\cos(50t-\pi/2)\ V$
c)
$V=625\cos(50t+5\pi/6)\ V$
d)
$V=-50\cos(50t+5\pi/6)=50\cos(50t-\pi/6)\ V$
Work Step by Step
given an inductor $L=250\ mH = 250\times 10^{-3}\ H$, we are required to find the voltages.
a) for current $i_c(t)=5\sin(25t)\ A$
the frequency is $\omega = 25\ Hz$
the impedance of the inductor $Z=j\omega L=j\times 25\times 250\times 10^{-3}=j6.25\ \Omega $
the voltage is
$V=I\times Z=(5\sin(25t))\times (j6.25)$
convert $sin$ to $cos$
$V=(5\cos(\pi/2-25t))\times (j6.25)$
using identity $cos(x)=cos(-x)$
$V=(5\cos(25t-\pi/2))\times (j6.25)$
convert to polar form
$V=(5\angle- 90^{\circ})\times (6.25\angle90^{\circ})$
(note: remember that $ j = 1 \angle90^{\circ}$)
$V=5\times 6.25 \ \angle(90-90)^{\circ}$
$V=31.25\ \angle0^{\circ}\ V$
convert back to Cartesian form and answer is
$$V=31.25\cos(25t)\ V$$
b) for current $i_c(t)=-10\cos(50t)\ A$
the frequency is $\omega = 50 \ Hz$
the impedance of the inductor $Z=j\omega L=j\times 50\times 250\times 10^{-3}=j12.5\ \Omega $
the voltage is
$$V=I\times Z=(-10\cos(50t))\times (j12.5)=-125\cos(50t+\pi/2)\ V$$
(note: $ j $ shifts a cosine by $90^{\circ}$)
c) for current $i_c(t)=25\cos(100t+\pi/3)\ A$
the frequency is $\omega = 100 \ Hz$
the impedance of the inductor $Z=j\omega L=j\times 100\times 250\times 10^{-3}=j25\ \Omega $
the voltage is
$V=I\times Z=(25\cos(100t+\pi/3))\times (j25)$
$$=625\cos(50t+5\pi/6)\ V$$
d) for current $i_c(t)=20\sin(10t-\pi/12)\ A$
the frequency is $\omega = 10 \ Hz$
the impedance of the inductor $Z=j\omega L=j\times 10\times 250\times 10^{-3}=j2.5\ \Omega $
the voltage is
$V=I\times Z=(20\cos(10t-\pi/12)))\times (j2.5)$
$=-50j\sin(50t+5\pi/6)\ V$
convert to $cos$
$=-50j\cos(\pi/2-(50t+5\pi/6))$
$=-50j\cos(-\pi/3-50t)$
use identity $cos(x)=cos(-x)$
$V=-50j\cos(50t+pi/3)$
$=-50\cos(50t+pi/3+pi/2)$
(note: $j$ shifts $cos$ by $+90^{\circ}$)
$$V=-50\cos(50t+5\pi/6)\ V$$
(note: even a minus sign shits a cosine by -180 so you can write the above answer as $50\cos(50t+5\pi/3-\pi)=50\cos(50t-\pi/6)\ V$ )
