We are tasked with writing a cosine function with the following characteristics:

1. Amplitude (A): -10
2. Period: $\frac{2\pi}{10}$
3. Phase Shift (ϕ): $\frac{-\pi}{6}$

We need to format the equation in the general form: $A \cos \left( \omega \left( x - \frac{\phi}{\omega} \right) \right)$ where:

• $A$ is the amplitude.
• $\omega$ is the angular frequency, related to the period by the formula $\omega = \frac{2\pi}{\text{Period}}$.
• $\phi$ is the phase shift.

Step-by-Step Solution:

1. Amplitude (A): Given as -10, so $A = -10$.

2. Period: The period is given as $\frac{2\pi}{10}$. The angular frequency $\omega$ is calculated by using the formula: $\omega = \frac{2\pi}{\text{Period}} = \frac{2\pi}{\frac{2\pi}{10}} = 10$

3. Phase Shift (ϕ): Given as $\frac{-\pi}{6}$.

Now, plugging these values into the cosine function formula:

$y = -10 \cos \left( 10 \left( x - \frac{\frac{-\pi}{6}}{10} \right) \right)$

Simplifying the phase shift expression:

$y = -10 \cos \left( 10 \left( x + \frac{\pi}{60} \right) \right)$

Final Answer:

$y = -10 \cos \left( 10 \left( x + \frac{\pi}{60} \right) \right)$

Would you like a more detailed breakdown or have any further questions?

Here are some related questions:

1. How do you calculate the phase shift for sinusoidal functions?
2. What is the significance of negative amplitude in a cosine function?
3. How does the period of a function relate to its angular frequency?
4. What are the differences between sine and cosine in terms of phase shift?
5. How do you graph a cosine function with a phase shift and amplitude change?

Tip: For functions with phase shifts, remember that $x - \phi$ means a shift to the right, while $x + \phi$ means a shift to the left.