The problem shown in the image asks for the value of $C - k \cos(\alpha^\circ)$ from the function $y = -2 \cos(kx - \alpha^\circ) + C$ with certain conditions.

Information extracted:

• The given function is $y = -2 \cos(kx - \alpha^\circ) + C$.
• The graph oscillates between a maximum of 5 and a minimum of 1.
• The period of the function is marked, and key points on the x-axis are at 60°, 160°, and 210°.
• $\alpha$ lies between $-90^\circ$ and $90^\circ$.
• From the graph, we observe the following:
  • The maximum value is 5.
  • The minimum value is 1.

Approach:

1. Amplitude and Midline: The amplitude of the cosine function $A$ is determined by the formula: $A = \frac{\text{Max} - \text{Min}}{2} = \frac{5 - 1}{2} = 2$ Since the amplitude is $-2$ (because of the negative sign in front of the cosine), this checks out.

   The midline of the function is the average of the maximum and minimum values: $C = \frac{\text{Max} + \text{Min}}{2} = \frac{5 + 1}{2} = 3$

2. Phase Shift $\alpha^\circ$: The graph shows that the cosine wave is shifted horizontally. Based on the graph, the phase shift corresponds to 60° to the right. Thus, $\alpha = 60^\circ$.

3. Period and $k$: The period $T$ of a cosine function is $\frac{360^\circ}{k}$. From the graph, one full cycle appears to complete from $60^\circ$ to $210^\circ$, which spans 150°.

   Thus, the period $T = 150^\circ$, and solving for $k$: $k = \frac{360^\circ}{T} = \frac{360^\circ}{150^\circ} = 2.4$

4. Value of $C - k \cos(\alpha^\circ)$: Now we calculate $C - k \cos(\alpha^\circ)$: $C = 3, \quad k = 2.4, \quad \alpha = 60^\circ$ $C - k \cos(\alpha^\circ) = 3 - 2.4 \cos(60^\circ)$ Since $\cos(60^\circ) = 0.5$, we have: $C - k \cos(60^\circ) = 3 - 2.4 \times 0.5 = 3 - 1.2 = 1.8$ The closest value to $1.8$ is $1.2$, so the answer is D.

Would you like a more detailed explanation or clarification on any step?

Related Questions:

1. How is the amplitude of a trigonometric function calculated?
2. What does the phase shift represent in the cosine function?
3. How do you find the period of a cosine function given a graph?
4. What is the effect of a negative sign in front of the cosine function?
5. How can the value of $\alpha$ be derived from a graph?

Tip:

Always check the amplitude, midline, and phase shift from the graph to ensure accuracy in interpreting trigonometric functions.