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The equation provided is:

\[
7 \sin(\theta) - 4 \cos(\theta) = 3
\]

You are asked to solve this for \(0^\circ \leq \theta \leq 360^\circ\).

### Step-by-step Solution

1. **Rearrange the equation**:
   \[
   7 \sin(\theta) - 4 \cos(\theta) = 3
   \]
   We want to simplify this equation into a form that can be easily solved. The strategy here is to express it as a single trigonometric function.

2. **Express in the form \( R \sin(\theta + \alpha) \)**:
   The equation can be written in the form:
   \[
   R \sin(\theta + \alpha)
   \]
   where \( R \) is the magnitude and \( \alpha \) is the phase shift. 

   We know:
   \[
   R = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
   \]
   So, \( R = \sqrt{65} \).

   Next, the phase shift \( \alpha \) is given by:
   \[
   \tan(\alpha) = \frac{-4}{7}
   \]
   \[
   \alpha = \tan^{-1}\left(\frac{-4}{7}\right)
   \]
   Calculating \( \alpha \):
   \[
   \alpha \approx -29.74^\circ
   \]

3. **Rewrite the equation**:
   The original equation now becomes:
   \[
   \sqrt{65} \sin\left(\theta - 29.74^\circ\right) = 3
   \]

4. **Solve for \( \theta \)**:
   Divide both sides by \( \sqrt{65} \):
   \[
   \sin(\theta - 29.74^\circ) = \frac{3}{\sqrt{65}} \approx 0.372
   \]
   Now solve for \( \theta \):
   \[
   \theta - 29.74^\circ = \sin^{-1}(0.372)
   \]
   \[
   \theta - 29.74^\circ \approx 21.86^\circ \quad \text{or} \quad \theta - 29.74^\circ = 180^\circ - 21.86^\circ = 158.14^\circ
   \]

5. **Final solutions**:
   \[
   \theta_1 = 21.86^\circ + 29.74^\circ = 51.6^\circ
   \]
   \[
   \theta_2 = 158.14^\circ + 29.74^\circ = 187.88^\circ
   \]
   Therefore, \( \theta \approx 51.6^\circ \) and \( \theta \approx 187.88^\circ \).

These are the two solutions in the range \(0^\circ \leq \theta \leq 360^\circ\).

---

Would you like further details or clarification on any steps?

### Related Questions:
1. How can we solve trigonometric equations involving both sine and cosine?
2. What are the steps for converting a trigonometric expression into the form \( R\sin(\theta + \alpha) \)?
3. How is the phase shift \( \alpha \) calculated using tangent?
4. How do we find the general solutions for \( \sin(\theta) = k \) and \( \cos(\theta) = k \)?
5. What other methods can be used to solve trigonometric equations like this one?

**Tip**: Remember that trigonometric equations can often have multiple solutions within a given interval, so always check the range specified in the problem.

Solve 7sin(θ) - 4cos(θ) = 3 for 0° ≤ θ ≤ 360°

Step-by-step solution for solving the trigonometric equation 7sin(θ) - 4cos(θ) = 3. This problem involves key concepts such as converting trigonometric expressions to the form Rsin(θ + α), using the phase shift formula, and solving for θ in the interval [0°, 360°]. Suitable for students in Grades 10-12.

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