Find the values of $$ \sin 18^{\circ}, \cos 18^{\circ} $$

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UpStudy AI · Accepted Answer

The exact values of $$ \sin 18^{\circ} $$ and $$ \cos 18^{\circ} $$ can be derived using geometric properties and trigonometric identities related to a regular pentagon. Here's how they are expressed:

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

$$
\cos 18^{\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{4} \quad \text{or equivalently} \quad \frac{\sqrt{5} + 1}{4} \sqrt{2 + \sqrt{5}}
$$

For clarity and simplicity, the cosine can also be written in a more compact form:

$$
\cos 18^{\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}
$$

**Summary:**
$$
\boxed{\,\sin 18^\circ = \frac{\sqrt{5}\,-\,1}{4} \quad\text{and}\quad \cos 18^\circ = \frac{\sqrt{10 + 2\,\sqrt{5}}}{4}\,}
$$
$$ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \quad \text{and} \quad \cos 18^\circ = \frac{\sqrt{10 + 2\sqrt{5}}}{4} $$

Find the values of \( \sin 18^{\circ}, \cos 18^{\circ} \)

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