We can simplify the expression $3\sin(18^\circ) - 4\sin^3(18^\circ)$ using trigonometric identities.

There's a well-known identity for this type of expression, related to the triple angle formula for sine:

$\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$

Setting $\theta = 18^\circ$, the expression becomes:

$3\sin(18^\circ) - 4\sin^3(18^\circ) = \sin(54^\circ)$

Now, using a calculator or known values for common angles, we know that:

$\sin(54^\circ) = 0.8090$

So, the value of $3\sin(18^\circ) - 4\sin^3(18^\circ)$ is approximately 0.8090.

Would you like further details on the derivation of this identity or any other clarification?

Here are 5 related questions to expand on the topic:

1. What is the derivation of the triple angle identity for sine?
2. How do we calculate the sine of special angles like $18^\circ$?
3. What is the general formula for $\sin(3\theta)$ and how is it used?
4. Can we use the same approach for other trigonometric functions, like cosine or tangent?
5. How does this identity relate to solving trigonometric equations?

Tip: Knowing special angle values and identities like the triple angle formula can help simplify trigonometric expressions efficiently.