Two harmonic oscillations are added, coinciding in direction and expressed by the equations

Two harmonic oscillations are added, coinciding in direction and expressed by the equations x1 = sin πt; x2 = cos πt. Determine the amplitude and the initial phase of the resulting oscillation, write its equation and give a vector diagram of the addition of the amplitudes.

Using the trigonometric reduction formula, we transform the second law:

x2 = cos (πt) = sin (π / 2 – πt).

We use the formula for adding two sines:

x1 + x2 = sin (πt) + sin (π / 2 – πt) = 2 * sin ((πt + π / 2 – πt) / 2) * cos ((πt – π / 2 + πt) / 2) = 2 * sin (π / 4) * cos (πt – π / 4) = √2 * cos (πt – π / 4) = √2 sin (3π / 4 – πt) = √2 * sin (πt + π / four).

This form describes the law of the resulting oscillation, it can be seen from it that the amplitude is equal to √2, and the initial phase is equal to π / 4.