The problem in the image involves analyzing the graph of a trigonometric function:

$y = -2 \cos(kx - \alpha^\circ) + C$

It asks for the value of $C - k \cos(\alpha^\circ)$, given the graph and some constraints such as $-90^\circ < \alpha < 90^\circ$.

Key observations from the graph:

• The graph shows a cosine wave.
• The amplitude appears to be 2, matching the coefficient of the cosine term in the equation.
• The graph crosses the Y-axis at around $y = 1$, which suggests that there is a vertical shift $C$, influencing the position of the cosine curve.
• The horizontal stretch and shift in the graph are influenced by the constants $k$ and $\alpha$.

Identifying key elements:

1. Amplitude: The coefficient of cosine is -2, so the amplitude is 2.

2. Period: From the graph, the function completes one cycle from $60^\circ$ to $210^\circ$, suggesting that the period is $210^\circ - 60^\circ = 150^\circ$. Using the formula for the period of a cosine function:

   $\text{Period} = \frac{360^\circ}{k}$ Equating the period to 150°:

   $\frac{360^\circ}{k} = 150^\circ \quad \Rightarrow \quad k = \frac{360^\circ}{150^\circ} = 2.4$

3. Vertical Shift (C): The vertical shift, $C$, can be inferred from the midline of the wave, which appears to be around $y = 1$, indicating that $C = 1$.

4. Phase Shift ($\alpha$): The graph is shifted to the right, starting at $60^\circ$. This indicates a phase shift $\alpha$, where $\alpha = 60^\circ$.

Final calculation:

The expression $C - k \cos(\alpha^\circ)$ becomes:

$C - k \cos(\alpha^\circ) = 1 - 2.4 \cos(60^\circ)$

Since $\cos(60^\circ) = 0.5$:

$C - k \cos(\alpha^\circ) = 1 - 2.4 \times 0.5 = 1 - 1.2 = -0.2$

Thus, the closest answer is B. -1.2.

Would you like further details on any specific step? Here are some related questions you might find helpful:

1. How is the amplitude of a cosine function determined from its equation?
2. What is the effect of the phase shift on the cosine graph?
3. How do you calculate the period of a trigonometric function?
4. What role does the vertical shift $C$ play in the graph of a trigonometric function?
5. How can you find the phase shift from the graph of a trigonometric function?

Tip: When analyzing the graph of a trigonometric function, always note the amplitude, period, vertical shift, and phase shift, as they directly correspond to the constants in the equation.