Combining sine & cosine

Japanese school year: Math II

What you learn

Learn how to combine linear expressions of sine and cosine into a single sinusoidal function. This technique is widely used to find extreme values and analyze combined simple harmonic oscillations. Having a solid understanding of addition formulas and unit circle representations will make mastering this method straightforward.

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Key points

This formula combines a sum of $\sin\theta$ and $\cos\theta$ into a single $\sin$ term. Here, $a$ and $b$ are coefficients; it is useful for finding maximum and minimum values or solving equations.

a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \alpha)

These equations determine the auxiliary angle $\alpha$ in the combination formula. If you plot the point $(a, b)$ on a plane, $\alpha$ is the angle between the positive $x$ -axis and the segment from the origin.

\cos\alpha = \frac{a}{\sqrt{a^2 + b^2}}, \quad \sin\alpha = \frac{b}{\sqrt{a^2 + b^2}}

This combination formula can only be applied when the angle $\theta$ is identical in both $\sin$ and $\cos$ . If the angles differ, you must first unify them using identities like double-angle formulas.

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Practice 1Coming soon