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Let's go through this question step by step.

### Part (i):
The task is to **expand \( \sin(2\theta + \theta) \)** to show that \( \sin(3\theta) = 3\sin\theta - 4\sin^3\theta \).

This requires using the **sum of angles identity** for sine:
\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]
For \( \sin(3\theta) = \sin(2\theta + \theta) \), apply this identity:
\[
\sin(3\theta) = \sin(2\theta)\cos(\theta) + \cos(2\theta)\sin(\theta)
\]

Next, use the **double angle formulas**:
- \( \sin(2\theta) = 2\sin\theta\cos\theta \)
- \( \cos(2\theta) = 1 - 2\sin^2\theta \)

Substitute these into the previous equation:
\[
\sin(3\theta) = (2\sin\theta\cos\theta)\cos\theta + (1 - 2\sin^2\theta)\sin\theta
\]
Simplify:
\[
\sin(3\theta) = 2\sin\theta\cos^2\theta + \sin\theta - 2\sin^3\theta
\]
Since \( \cos^2\theta = 1 - \sin^2\theta \), substitute this:
\[
\sin(3\theta) = 2\sin\theta(1 - \sin^2\theta) + \sin\theta - 2\sin^3\theta
\]
Expand and simplify:
\[
\sin(3\theta) = 2\sin\theta - 2\sin^3\theta + \sin\theta - 2\sin^3\theta
\]
\[
\sin(3\theta) = 3\sin\theta - 4\sin^3\theta
\]

Thus, part (i) is shown.

---

### Part (ii):
You need to show that after substituting \( x = \frac{2\sin\theta}{\sqrt{3}} \), the equation \( x^3 - x + \frac{1}{6}\sqrt{3} = 0 \) can be rewritten in the form \( \sin(3\theta) = \frac{3}{4} \).

First, express \( \sin\theta \) in terms of \( x \) by rearranging the substitution:
\[
\sin\theta = \frac{x\sqrt{3}}{2}
\]

Now substitute this expression for \( \sin\theta \) into the equation \( \sin(3\theta) = 3\sin\theta - 4\sin^3\theta \):
\[
\sin(3\theta) = 3\left( \frac{x\sqrt{3}}{2} \right) - 4\left( \frac{x\sqrt{3}}{2} \right)^3
\]
Simplify the powers of \( x \):
\[
\sin(3\theta) = \frac{3x\sqrt{3}}{2} - 4\left( \frac{x^3 3\sqrt{3}}{8} \right)
\]
This becomes:
\[
\sin(3\theta) = \frac{3\sqrt{3}}{2}x - \frac{3\sqrt{3}}{2}x^3
\]
Now multiply both sides by \( \frac{2}{3\sqrt{3}} \) to isolate terms:
\[
x^3 - x + \frac{1}{6}\sqrt3

By first expanding sin(2θ + θ), show that sin(3θ) = 3sin(θ) - 4sin³(θ). Show that, after making the substitution x = (2sin(θ))/√3, the equation x³ - x + (1/6)√3 = 0 can be written in the form sin(3θ) = 3/4. Hence solve the equation x³ - x + (1/6)√3 = 0, giving your answers correct to 3 significant figures.

Learn how to expand sin(2θ + θ) using trigonometric identities to derive the expression sin(3θ) = 3sin(θ) - 4sin³(θ). Additionally, explore the substitution method to solve the cubic equation x³ - x + (1/6)√3 = 0.

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