Question 10

Question 10

For real numbers $a$ and $b$, where $a \neq 0$ and $b \neq 0$, we can find numbers $\alpha, \beta, \gamma, \delta$ and $R$ such that $a \cos{x} + b\sin{x}$ can be written in the following 4 forms:

$$\begin{aligned} R\sin{\left(x + \alpha\right)} \\ R\sin{\left(x - \beta\right)} \\ R\cos{\left(x + \gamma\right)} \\ R\cos{\left(x - \delta\right)} \\ \end{aligned}$$

where $R > 0$ and $0 < \alpha, \beta, \gamma, \delta < 2\pi$.

What is the value of $\alpha + \beta + \gamma + \delta$?

A. $0$

B. $\pi$

C. $2\pi$

D. $4\pi$

Solution

If we have $0 < \alpha < 2\pi$ then

$$\begin{aligned} R\sin{\left(x + \alpha\right)} \\ R\sin{\left(x + \alpha - 2\pi\right)} \\ R\sin{\left(x - \left(2\pi - \alpha\right)\right)} \end{aligned}$$

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