Question

If \( i_1 = 3 \sin(\omega t) \) and \( i_2 = 4 \cos(\omega t) \), then \( i_3 \) is

• A. \( 5 \sin(\omega t + 53^\circ) \)
• B. \( 5 \sin(\omega t + 37^\circ) \)
• C. \( 6 \sin(\omega t + 45^\circ) \)
• D. \( 5 \cos(\omega t + 53^\circ) \)

Hint:

The resultant of two sinusoidal currents can be found using the Pythagorean theorem for amplitudes and the tangent of the phase difference.

Answer 1

The Correct Option is A

Step 1: Superposition of Currents.
To find the resultant current \( i_3 \), we use the vector addition of the sinusoidal currents. The amplitude of the resultant current is: \[ i_3 = \sqrt{i_1^2 + i_2^2} \] The phase angle \( \phi \) is given by: \[ \tan(\phi) = \frac{4}{3} \] Thus, \( \phi = 53^\circ \).
Step 2: Conclusion.
The correct answer is (A), \( 5 \sin(\omega t + 53^\circ) \).