We are given the equation:

\[
165 + 140 \sin(\alpha) - 225 \cos(\alpha) = 0
\]

To solve this trigonometric equation for \(\alpha\), let's proceed step by step.

### Step 1: Rearrange the equation
We can isolate the trigonometric terms by moving the constant to the other side:

\[
140 \sin(\alpha) - 225 \cos(\alpha) = -165
\]

### Step 2: Divide through by the square root of the sum of squares of the coefficients
To simplify, recognize that this equation is of the form \( A \sin(\alpha) + B \cos(\alpha) = C \). We can divide both sides of the equation by \(\sqrt{140^2 + (-225)^2}\) to express it in a simpler form:

First, calculate \(\sqrt{140^2 + 225^2}\):

\[
\sqrt{140^2 + 225^2} = \sqrt{19600 + 50625} = \sqrt{70225} = 265
\]

Now divide both sides of the equation by 265:

\[
\frac{140}{265} \sin(\alpha) - \frac{225}{265} \cos(\alpha) = \frac{-165}{265}
\]

Simplifying:

\[
0.528 \sin(\alpha) - 0.849 \cos(\alpha) = -0.623
\]

### Step 3: Express the equation in terms of a single trigonometric function
We can express this equation as:

\[
\sin(\alpha - \theta) = -0.623
\]

where \(\theta\) is the angle such that:

\[
\cos(\theta) = \frac{225}{265} = 0.849 \quad \text{and} \quad \sin(\theta) = \frac{140}{265} = 0.528
\]

Thus, \(\theta = \tan^{-1}\left(\frac{140}{225}\right)\).

### Step 4: Solve for \(\alpha\)
Now we solve \(\sin(\alpha - \theta) = -0.623\). The general solution for \(\sin(x) = -y\) is:

\[
\alpha - \theta = \sin^{-1}(-0.623) + 2k\pi \quad \text{or} \quad \alpha - \theta = \pi - \sin^{-1}(-0.623) + 2k\pi
\]

Calculating \(\sin^{-1}(-0.623)\), we get approximately:

\[
\sin^{-1}(-0.623) \approx -0.672 \text{ radians}
\]

Thus:

\[
\alpha - \theta = -0.672 \quad \text{or} \quad \alpha - \theta = \pi + 0.672
\]

Now, add \(\theta\) to both sides to find \(\alpha\).

Would you like further explanation or details on any specific steps?

Here are some related questions to deepen your understanding:

1. What are some methods to solve general trigonometric equations like this?
2. How can you solve similar problems with different trigonometric functions (e.g., using \(\tan\))?
3. What is the geometric interpretation of solving \(\sin(\alpha - \theta)\)?
4. How can you simplify trigonometric equations using identities?
5. How does the unit circle help in solving \(\sin^{-1}\) and \(\cos^{-1}\)?

**Tip:** When solving trigonometric equations, always check for multiple solutions within the given interval, as trig functions are periodic.

Learn how to solve the trigonometric equation 165 + 140sin(α) − 225cos(α) = 0 step by step. This problem involves key trigonometric concepts, including rearranging trigonometric equations and solving for the angle α using the inverse sine theorem. Suitable for students in Grades 11-12.

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