Question

Show that acos(wt) bsin(wt) can be written in the form ________.

Answer 1

The expression a cos(wt) + b sin(wt) can be rewritten using the identity R cos(wt + α) by finding the amplitude R and phase angle α to match the coefficients a and b.

Step-by-step explanation:

To show that the expression a cos(wt) + b sin(wt) can be written in a different form, we can use a trigonometric identity. We start by recognizing that any cosine or sine function with a phase shift can be represented as a combination of cosine and sine functions. Specifically, a general form R cos(wt + α) can be used, where R is the amplitude and α is the phase angle.

Using the cosine addition formula, cos(C + D) = cos C cos D - sin C sin D, we can express R cos(wt + α) as R (cos α cos(wt) - sin α sin(wt)). If we match coefficients, we can find that a = R cos α and b = -R sin α. Solving these gives us the values for R and α.

Therefore, a cos(wt) + b sin(wt) can be written as R cos(wt + α), where R = √(a² + b²) and α = atan2(-b, a).