To demonstrate that $f(x) = -\frac{1}{2} \sin(4x) - \frac{\sqrt{3}}{2} \cos(4x)$ is a sinusoidal function, we can rewrite it in the form $S(x) = A \sin(\omega x + \varphi) + B$, where $A$, $\omega$, $\varphi$, and $B$ are constants.

Step 1: Identify the Parameters

The given function is: $f(x) = -\frac{1}{2} \sin(4x) - \frac{\sqrt{3}}{2} \cos(4x)$ We can observe that this is a linear combination of $\sin(4x)$ and $\cos(4x)$ with coefficients $-\frac{1}{2}$ and $-\frac{\sqrt{3}}{2}$, respectively.

Step 2: Rewrite the Function in Terms of a Single Sinusoidal Function

To rewrite the expression as a single sinusoidal function, we can use the identity: $a \sin(4x) + b \cos(4x) = R \sin(4x + \varphi)$ where:

• $R = \sqrt{a^2 + b^2}$ (the amplitude),
• $\varphi = \arctan\left(\frac{b}{a}\right)$ (the phase shift).

In this case:

• $a = -\frac{1}{2}$
• $b = -\frac{\sqrt{3}}{2}$

Calculating $R$:

$R = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2}$ $R = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$

Thus, $R = 1$.

Calculating $\varphi$:

$\varphi = \arctan\left(\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \arctan\left(\frac{\sqrt{3}}{1}\right) = \arctan(\sqrt{3})$ Since $\arctan(\sqrt{3}) = \frac{\pi}{3}$, we have $\varphi = \frac{\pi}{3}$.

Step 3: Substitute Back

Now we can rewrite $f(x)$ as: $f(x) = R \sin(4x + \varphi)$ Substituting $R = 1$ and $\varphi = \frac{\pi}{3}$: $f(x) = \sin\left(4x + \frac{\pi}{3}\right)$

Final Answer

Thus, the function can be written as: $f(x) = \sin\left(4x + \frac{\pi}{3}\right)$ which is in the form $S(x) = A \sin(\omega x + \varphi) + B$ with:

• $A = 1$
• $\omega = 4$
• $\varphi = \frac{\pi}{3}$
• $B = 0$

This confirms that $f(x)$ is indeed a sinusoidal function.

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Would you like more details on any part of the solution?

Related Questions:

1. How do you determine the amplitude and phase shift of a sinusoidal function?
2. Can you explain why any linear combination of sine and cosine functions can be written as a single sinusoid?
3. How do we convert a cosine function to an equivalent sine function?
4. What is the effect of different values of $A$ and $\omega$ on the graph of a sinusoidal function?
5. How does the phase shift affect the graph of the sinusoidal function?

Tip:

Remember, combining $a \sin(x) + b \cos(x)$ into a single sinusoidal term often simplifies analysis, especially when graphing or solving equations.