The equation is given as:

\[
8 \sin(\alpha) - 15 \cos(\alpha) = 17
\]

To solve this, we use the following steps:

### Step 1: Convert the equation into a single trigonometric function
We express this equation in the form \( R \sin(\alpha + \theta) \), where:

\[
R = \sqrt{8^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\]

Now, using the identity:

\[
R \sin(\alpha + \theta) = 8 \sin(\alpha) - 15 \cos(\alpha)
\]

we need to find \(\theta\) such that:

\[
\cos(\theta) = \frac{8}{17} \quad \text{and} \quad \sin(\theta) = \frac{-15}{17}
\]

Thus, the equation becomes:

\[
17 \sin(\alpha + \theta) = 17
\]

### Step 2: Simplify the equation
Now divide both sides by 17:

\[
\sin(\alpha + \theta) = 1
\]

### Step 3: Solve for \(\alpha + \theta\)
The general solution to \(\sin(\alpha + \theta) = 1\) is:

\[
\alpha + \theta = \frac{\pi}{2} + 2n\pi \quad \text{where} \quad n \in \mathbb{Z}
\]

Now, we calculate \(\theta\). Since:

\[
\tan(\theta) = \frac{-15}{8}
\]

We find:

\[
\theta = \tan^{-1}\left(\frac{-15}{8}\right)
\]

Now you can substitute \(\theta\) back to find the value of \(\alpha\).

Would you like further clarification or more details?

Here are 5 related questions:
1. How is the identity \(R \sin(\alpha + \theta)\) derived?
2. Can this method be applied to other forms of trigonometric equations?
3. What if the equation had a different constant on the right-hand side?
4. How do you determine the value of \(\theta\) in general?
5. What are the general solutions for other trigonometric equations?

**Tip:** Always look for a way to simplify complex trigonometric equations using known identities.

Learn how to solve the trigonometric equation 8sin(α) - 15cos(α) = 17 using angle addition identities and trigonometric properties. This solution covers key trigonometric concepts and is suitable for students in Grades 10-12.

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