Solve the following equation for all values of 0, is less than or equal to, theta, is less than, 2, pi0≤θ<2π.
3, tangent, squared, theta, plus, 3, tangent, theta, equals, 0
3tan 
2
 θ+3tanθ=0

The given equation is:

\[
3 \tan^2 \theta + 3 \tan \theta = 0
\]

To solve this equation, let's follow these steps:

### Step 1: Factor the Equation
First, factor out the common term:

\[
3 \tan \theta (\tan \theta + 1) = 0
\]

This equation is satisfied when either of the factors is zero:

\[
3 \tan \theta = 0 \quad \text{or} \quad \tan \theta + 1 = 0
\]

### Step 2: Solve Each Factor

1. **For \( 3 \tan \theta = 0 \):**

   \[
   \tan \theta = 0
   \]

   The tangent function is zero at:

   \[
   \theta = 0, \pi
   \]

2. **For \( \tan \theta + 1 = 0 \):**

   \[
   \tan \theta = -1
   \]

   The tangent function equals \(-1\) at:

   \[
   \theta = \frac{3\pi}{4}, \frac{7\pi}{4}
   \]

### Step 3: List All Solutions Within the Given Interval

Combining all solutions, the values of \(\theta\) that satisfy the equation \( 0 \leq \theta < 2\pi \) are:

\[
\theta = 0, \pi, \frac{3\pi}{4}, \frac{7\pi}{4}
\]

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions you might be interested in:
1. How do you solve equations involving other trigonometric functions like sine or cosine?
2. What are the properties of the tangent function, especially its periodicity?
3. How can we verify solutions of trigonometric equations graphically?
4. How do you solve trigonometric equations with multiple angles, such as \( \tan(2\theta) \)?
5. What are some common techniques for factoring trigonometric expressions?

**Tip:** Remember that the tangent function has a period of \(\pi\), which means solutions repeat every \(\pi\) units.

Solve the following equation for all values of 0 ≤ θ < 2π: 3tan^2(θ) + 3tan(θ) = 0

This guide provides a step-by-step solution for the trigonometric equation 3tan^2(θ) + 3tan(θ) = 0 within the interval 0 ≤ θ < 2π. Learn how to factor and solve for θ, including solutions at θ = 0, π, 3π/4, and 7π/4.

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