Solve the following trigonometric equation assuming that 0 ≤ 𝜃 < 2𝜋. 2 − 3 sin(𝜃) = 1 − 5 sin(𝜃)

To solve the trigonometric equation \( 2 - 3 \sin(\theta) = 1 - 5 \sin(\theta) \), follow these steps:

1. **Start by isolating the sine terms** on one side of the equation:

\[
2 - 1 = -5 \sin(\theta) + 3 \sin(\theta)
\]

2. **Simplify both sides**:

\[
1 = -2 \sin(\theta)
\]

3. **Solve for** \( \sin(\theta) \):

\[
\sin(\theta) = -\frac{1}{2}
\]

4. **Determine the general solutions** for \( \sin(\theta) = -\frac{1}{2} \). The sine function equals \(-\frac{1}{2}\) at:

\[
\theta = \frac{7\pi}{6} \quad \text{and} \quad \theta = \frac{11\pi}{6}
\]

These are the solutions within the interval \( 0 \leq \theta < 2\pi \).

**Final Answer:**
\[
\theta = \frac{7\pi}{6} \quad \text{and} \quad \theta = \frac{11\pi}{6}
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would you solve the equation \( 3 \sin(\theta) - 2 = 5 \sin(\theta) - 4 \)?
2. What are the solutions for \( \sin(\theta) = \frac{1}{2} \) within the interval \( 0 \leq \theta < 2\pi \)?
3. How do you solve the equation \( 2 \cos(\theta) + 3 = 4 \sin(\theta) \)?
4. What are the general solutions for \( \cos(\theta) = \frac{1}{2} \)?
5. How do you solve a trigonometric equation involving both sine and cosine, like \( \sin(\theta) + \cos(\theta) = 1 \)?

**Tip:** When solving trigonometric equations, it's often helpful to use identities or known values of sine and cosine for specific angles to simplify the problem.

Learn how to solve the trigonometric equation 2 - 3 sin(𝜃) = 1 - 5 sin(𝜃) step-by-step. This problem involves trigonometric identities and equations, focusing on properties of the sine function. Suitable for high school students in Grades 10-12.

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