Question

In the equation of alternating current is given by \[E = 158\sin \left( {200\pi t} \right){\text{V}}\]. The value of voltage at time \[t = 1/400s\] is
A. 168 V
B. -79 V
C. 79 V
D. 158 V

Answer 1

Hint: Since the unit of the equation is in volt, it signifies that the voltage of the alternate current (alternate current electricity) was given. Substitute the value of the time given for the variable \[t\] in the equation of the voltage given.

Complete step by step answer:
We said that the equation of an alternate current is \[E = 158\sin \left( {200\pi t} \right){\text{V}}\]. With the quantity being in volts, the quantity given to us governed by the equation above is the voltage of the electricity.
We are asked to find the value of the voltage after time \[t = 1/400s\].
To do this, we simply need to substitute the given value of time into the equation given above, as in:
\[E = 158\sin \left( {200\pi \dfrac{1}{{400}}} \right){\text{V}}\]
Hence, by dividing, we have that
\[E = 158\sin \left( {\dfrac{\pi }{2}} \right){\text{V}}\]
Pi over 2 is equivalent to 90 degrees. And the sine of 90 degrees is equal to one.
Hence, by carrying out the operation, we have
\[E = 158\left( 1 \right){\text{V}}\]
Hence, the correct option is 158 V. As seen, this corresponds to option D.
Hence, the correct answer is option D.

Note: For clarity, the fact that pi over 2 (unit in radians) is the same as 90 degrees can be gotten from the conversion relation. Generally, we have
\[{A_d} = {A_r} \times \dfrac{{180}}{\pi }\] where \[{A_d}\] is angle in degree, and \[{A_r}\] is angle in radian.
Hence, we have \[{A_d} = \dfrac{\pi }{2} \times \dfrac{{180}}{\pi } = 90^\circ \].
Note also, that angles gotten from SI unit quantities are in radians. Essentially, in scientific calculations of angles, the unit is in radians. Usually, however, the rad (as radian is often shortened) does not appear explicitly in many cases, and even when the angle is being found the result might seem unit-less. Instead, we just note that for angles, the unit is the radian or rad.