Combining Trigonometric Sums

When we have a sum of the form $a\sin{\theta} + b\cos{\theta}$ , we are combining two periodic functions with the same period - $2\pi$ - so we would expect the sum to also be a periodic function with a period of $2\pi$ .

In fact, it turns out that with a little bit of algebraic rearrangement, we are able to write these sums in the form $R\cos{\left(\theta \pm \alpha\right)}$ or $R\sin{\left(\theta \pm \alpha\right)}$ where $\alpha$ is some angle between $-\pi$ and $\pi$ .

We will do this by re-writing $a$ and $b$ into a form like $R\cos{\alpha}$ and $R\sin{\alpha}$ , and then we can use a compound angle formula to simplify it.

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