Determine the amplitude, period, and phase shift of $f(t) = 2 sin t + \sqrt{3}cos t$

Question

For the given : $f(t) = 2sint + \sqrt{3}cost$, we are tasked to find the amplitude, period and phase shift.

From the book :

• $f(t) = 2sint + \sqrt{3}cost$ = a sin (t + c)
• $a = \sqrt{2^2 + (\sqrt{3})^2} = 4$
• $tan c = \frac{\sqrt{3}}{2} $ ; $ c = tan^{-1}({\frac{\sqrt{3}}{2}}) = \frac{\pi}{3}$
• $f(t) = 2sint + \sqrt{3}cost$ = $4 sin (t + \frac{\pi}{3})$

Final Answer : $ f(t) =4 sin (t + \frac{\pi}{3})$ where A = 4, period = $2*\pi$ and phase shift = $\frac{\pi}{3} $ to the left.

My questions :

1. On the first bullet, how did he know that the given function is equal to $a sin(t + c)$ by looking at the equation f(t) = sin + cos?
2. On the 3rd bullet, $tanc = \frac{\sqrt{3}}{2}$; Why is the $\sqrt{3}$ on the numerator and 2 on the denominator? Tan is equal to $\frac{sin}{cos}$, so from the given $f(t) = 2sint + \sqrt{3}cost$, why is the $\sqrt{3}$ the value for sine (numerator) when it is in the term containing cosine?
3. On bullet 3 & 4, why is the value of c = $\frac{\pi}{3}$? I mean, yeah the c is solved via the preceding formula but why is that the formula for the c? The value that will determine the phase shift of the function?

Note :

The answer on the book maybe wrong as the amplitude must be equal to $\sqrt{7}$ and $c = tan^{-1}({\frac{\sqrt{3}}{2}})$ is not equal to $ \frac{\pi}{3}$. But even though the answer is wrong, the method of solving is right. (I think.. based on the other problems.)

Again, thank you in advance. Any help would be appreciated.

Answer 1

This is achieved through the linear combination of $\sin$ and $\cos$:

$$a\sin x + b\cos x = c\sin(x + \varphi)$$

Where:

$$c = \sqrt{a^2 + b^2}$$ $$\varphi = \text{atan2}(b, a)$$

In this case $\text{atan2}(b, a)$ represents a generalized $\arctan(b/a)$. Thus:

$$c = \sqrt{2^2 + \sqrt{3}^2} = \sqrt 7\neq 4$$ $$\varphi = \arctan \frac {\sqrt3}2 \implies \tan \varphi = \frac {\sqrt3} 2$$

Notice that $\arctan(\sqrt 3/2)$ is not $\arctan \sqrt 3$ so it is not $\pi/3$. It is approximately $0.714 \text{ rad}$:

$$2\sin t + \sqrt 3 \cos t \approx \sqrt 7 \sin(t + 0.714)$$

I think you'll need to get a new book if it's making such blatant errors.