Trigonometric Identities: Real-World Scenarios (6)

Question

A pendulum’s projected horizontal motion follows $x(t) = 5\sin(2t) + 12\cos(2t)$. Express this as a single sinusoid $R\sin(2t + \phi)$ and find the maximum displacement.

Solution — Step by Step

Why This Works

The harmonic addition formula uses the geometric fact that any linear combination of $\sin$ and $\cos$ at the same frequency is itself a sinusoid at that frequency, just shifted in phase.

Geometrically: $a\sin + b\cos$ is the dot product of $(a, b)$ with $(\sin\theta, \cos\theta)$, which traces a circle of radius 1. The maximum dot product is $\sqrt{a^2 + b^2}$, achieved when $(a, b)$ aligns with the unit vector.

Alternative Method — Use Cosine Addition

We could also write $a\sin\theta + b\cos\theta = R\cos(\theta - \alpha)$ where $\tan\alpha = a/b$.

Both forms give the same amplitude $\sqrt{a^2 + b^2}$ — only the phase representation changes. Use whichever the question asks for.

Common Mistake

Students often compute $R = a + b$ instead of $\sqrt{a^2 + b^2}$, getting $5 + 12 = 17$ instead of $13$. The amplitudes don’t add linearly — they add in quadrature (Pythagoras).

This formula has direct physics applications: combining two waves at the same frequency, AC circuit phasor sums, and resolving force components.