Compute $\sin^3 18^\circ + \sin^2 18^\circ.$

We need to compute the value of $\sin^3 18^\circ + \sin^2 18^\circ$.

### Step 1: Let $ y = \sin 18^\circ $
Define $ y = \sin 18^\circ $. Then, we want to find:

\[
y^3 + y^2
\]

### Step 2: Use the known value of $\sin 18^\circ$
Recall that the exact value of $\sin 18^\circ$ is:

\[
\sin 18^\circ = \frac{\sqrt{5} - 1}{4}
\]

So, $ y = \frac{\sqrt{5} - 1}{4} $.

### Step 3: Calculate $ y^2 $
First, we square $ y $:

\[
y^2 = \left(\frac{\sqrt{5} - 1}{4}\right)^2 = \frac{(\sqrt{5} - 1)^2}{16}
\]

Expanding the square:

\[
(\sqrt{5} - 1)^2 = (\sqrt{5})^2 - 2 \times \sqrt{5} \times 1 + 1^2 = 5 - 2\sqrt{5} + 1 = 6 - 2\sqrt{5}
\]

So,

\[
y^2 = \frac{6 - 2\sqrt{5}}{16} = \frac{3 - \sqrt{5}}{8}
\]

### Step 4: Calculate $ y^3 $
Now we compute $ y^3 $:

\[
y^3 = y \times y^2 = \frac{\sqrt{5} - 1}{4} \times \frac{3 - \sqrt{5}}{8} = \frac{(\sqrt{5} - 1)(3 - \sqrt{5})}{32}
\]

Expanding the product:

\[
(\sqrt{5} - 1)(3 - \sqrt{5}) = \sqrt{5} \times 3 - \sqrt{5} \times \sqrt{5} - 1 \times 3 + 1 \times \sqrt{5} = 3\sqrt{5} - 5 - 3 + \sqrt{5} = 4\sqrt{5} - 8
\]

Thus,

\[
y^3 = \frac{4\sqrt{5} - 8}{32} = \frac{\sqrt{5} - 2}{8}
\]

### Step 5: Add $ y^3 $ and $ y^2 $
Finally, we add $ y^3 $ and $ y^2 $:

\[
y^3 + y^2 = \frac{\sqrt{5} - 2}{8} + \frac{3 - \sqrt{5}}{8} = \frac{\sqrt{5} - 2 + 3 - \sqrt{5}}{8} = \frac{1}{8}
\]

Thus, the value of $\sin^3 18^\circ + \sin^2 18^\circ$ is:

\[
\boxed{\frac{1}{8}}
\]

Do you have any questions, or would you like further details?

Here are five related questions you might find helpful:
1. What is the derivation of $\sin 18^\circ = \frac{\sqrt{5} - 1}{4}$?
2. How can the value of $\cos 18^\circ$ be determined from $\sin 18^\circ$?
3. How can trigonometric identities help simplify expressions involving cubes and squares of trigonometric functions?
4. How does the sine function behave for other angles that are fractions of 90° or 180°?
5. How can trigonometric values be used to solve algebraic equations involving roots?

**Tip:** Familiarity with exact trigonometric values for common angles (like $18^\circ$, $36^\circ$, etc.) can simplify complex expressions and help in solving related problems efficiently.

Learn how to compute $\sin^3 18^\circ + \sin^2 18^\circ$ with a step-by-step solution. Understand the trigonometric identity involved and the exact value of $\sin 18^\circ$. Suitable for high school students studying trigonometry.

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