We are asked to express $-15\sin{x} + 8\cos{x}$ in the form $k\sin(x + \alpha)$, where $0 \leq \alpha < 2\pi$. This transformation is based on the idea of combining sine and cosine terms into a single sinusoidal expression. Here's how we can proceed:

Step-by-Step Process:

1. Expression to Match: We are trying to match the given expression $-15\sin{x} + 8\cos{x}$ to the form: $k\sin(x + \alpha) = k(\sin{x}\cos{\alpha} + \cos{x}\sin{\alpha}).$ Expanding the right-hand side: $k(\sin{x}\cos{\alpha} + \cos{x}\sin{\alpha}) = k\cos{\alpha}\sin{x} + k\sin{\alpha}\cos{x}.$

2. Coefficient Matching: From this, we equate the coefficients of $\sin{x}$ and $\cos{x}$ in both expressions: -15 = k\cos{\alpha} \quad \text{(for the \sin{x} term)}, 8 = k\sin{\alpha} \quad \text{(for the \cos{x} term)}.

3. Find $k$: The amplitude $k$ is given by: $k = \sqrt{(-15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17.$

4. Find $\alpha$: To find $\alpha$, use the relationships: $\cos{\alpha} = \frac{-15}{17}, \quad \sin{\alpha} = \frac{8}{17}.$ Now, compute $\alpha$ using the arctangent function: $\alpha = \tan^{-1}\left(\frac{8}{-15}\right).$ The angle $\alpha$ is in the second quadrant because $\cos{\alpha}$ is negative and $\sin{\alpha}$ is positive.

   $\alpha = \tan^{-1}\left(\frac{8}{-15}\right) \approx \tan^{-1}(-0.5333).$ Using a calculator, $\alpha \approx 2.654$ radians (to 3 decimal places).

Final Answer:

The given expression $-15\sin{x} + 8\cos{x}$ can be written as: $17\sin(x + 2.654).$

Let me know if you'd like further details or clarifications!

Related Questions:

1. How is the amplitude $k$ derived from the coefficients of $\sin{x}$ and $\cos{x}$?
2. Why does the angle $\alpha$ end up in the second quadrant?
3. Can we use a different method to express $-15\sin{x} + 8\cos{x}$?
4. How can we verify that the expression $17\sin(x + 2.654)$ is equivalent to the original one?
5. What would the expression look like if we used cosine instead of sine?

Tip:

To remember which quadrant an angle lies in when using inverse trigonometric functions, pay attention to the signs of sine and cosine!