The image shows a trigonometric function problem. The function is given as $y = -2 \cos(kx - \alpha)$, and the problem appears to ask for the values of $C - K \cos 5\alpha$.

Steps to solve the problem:

1. Identify the amplitude, period, and phase shift:

   • The general form $y = A \cos(kx - \alpha)$ is a cosine function with:
     • $A = -2$: The amplitude (absolute value of $A$) is 2, and since it's negative, the graph is reflected vertically.
     • $k$: The angular frequency affects the period, which is $\frac{2\pi}{k}$.
     • $\alpha$: This is the phase shift (in degrees or radians).

2. Amplitude: From the graph, the maximum value is 5 and the minimum is 1. This suggests that the amplitude of the function has been shifted vertically.

3. Period: From the graph, the distance between two crests (one full period) seems to be from $60^\circ$ to $210^\circ$, meaning the period of the function is $210^\circ - 60^\circ = 150^\circ$. Therefore: $\text{Period} = \frac{360^\circ}{k} = 150^\circ$ Solving for $k$: $k = \frac{360^\circ}{150^\circ} = 2.4$

4. Phase Shift: The phase shift is given by $\alpha$, which shifts the graph horizontally. From the graph, it seems the function begins at $60^\circ$, so $\alpha = 60^\circ$.

5. Vertical Shift: Given that the graph's maximum is at 5 and minimum is at 1, the function has been shifted upwards by 3 units.

Thus, the equation of the function should be: $y = -2 \cos(2.4x - 60^\circ) + 3$

6. Final step: Solve for $C - K \cos(5\alpha)$:

   • Here, $C$ likely refers to the vertical shift, which is 3.
   • $K = 2$ (from the amplitude), and $\alpha = 60^\circ$.

   So: $C - K \cos(5\alpha) = 3 - 2 \cos(5 \times 60^\circ) = 3 - 2 \cos(300^\circ)$ Using $\cos(300^\circ) = 0.5$, we get: $C - K \cos(5\alpha) = 3 - 2 \times 0.5 = 3 - 1 = 2$

Thus, the value of $C - K \cos(5\alpha)$ is 2.

Do you want more details or have any questions?

Here are 5 related questions:

1. How does the value of $k$ affect the period of the cosine function?
2. What is the significance of a vertical shift in trigonometric functions?
3. How do you calculate the phase shift for a cosine function?
4. Why is the amplitude taken as the absolute value of $A$?
5. What is the relationship between angular frequency and the period of a function?

Tip: For trigonometric functions, the period is always calculated as $\frac{2\pi}{k}$ (in radians) or $\frac{360^\circ}{k}$ (in degrees).