PhysLab.net – Trig Identity for cos(t) + sin(t)

This trig identity shows that a combination of sine and cosine functions can be written as a single sine function with a phase shift.
a cos(t) + b sin(t) = $\sqrt{a^2 + b^2} \; \sin(t + \tan^{-1} \frac{a}{b})$
for b ≠ 0 and − ^π⁄₂ < tan⁻¹ ^a⁄_b < ^π⁄₂ (Note that tan⁻¹ means arctan.) The phase shift is the quantity tan⁻¹(^a⁄_b), it has the effect of shifting the graph of the sine function to the left or right. To derive this trig identity, we presume that the combination a cos(t) + b sin(t) can be written in the form c sin(K + t) for unknown constants c, K. a cos(t) + b sin(t) = c sin(K + t)
a cos(t) + b sin(t) = c sin(K) cos(t) + c cos(K) sin(t) We used the formula for sine of a sum of angles to expand the right hand side above. To have equality for any value of t, the coefficients of cos(t) and sin(t) must be equal on the left and right sides of the equation. a = c sin(K)
b = c cos(K) Solving this system of simultaneous equations leads us to c = ± √(a² + b²)
K = tan⁻¹ ^a⁄_b So the trig identity for b ≠ 0 is a cos(t) + b sin(t) = ± √(a²+b²) sin(t + tan⁻¹ ^a⁄_b) If we limit the arctan to be within − ^π⁄₂ < tan⁻¹ ^a⁄_b < ^π⁄₂ then we can always choose the + in front of the square root.