The expression is \( \cos\left(\frac{2}{5} \pi\right) \). To solve this, we need to evaluate the cosine function at this angle.

Given that:

\[
\cos\left(\frac{2}{5} \pi\right)
\]

### Step-by-Step Solution:

1. **Convert the Angle to Degrees:**
   \[
   \frac{2}{5} \pi \text{ radians} = \frac{2}{5} \times 180^\circ = 72^\circ
   \]

2. **Cosine of 72 Degrees:**
   \( 72^\circ \) is not a standard angle where the exact value is commonly known, but it can be expressed using trigonometric identities. 

   Alternatively, the exact value can be derived from the cosine of 36°, as \( \cos(72^\circ) = \sin(18^\circ) \), and \( \sin(18^\circ) \) is known to be:
   \[
   \sin(18^\circ) = \frac{\sqrt{5}-1}{4}
   \]

   Thus:
   \[
   \cos\left(\frac{2}{5} \pi\right) = \frac{\sqrt{5}-1}{4}
   \]

So the exact value is:

\[
\cos\left(\frac{2}{5} \pi\right) = \frac{\sqrt{5}-1}{4}
\]

Would you like more details or have any questions?

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Here are some related questions:
1. How is the cosine of other non-standard angles derived?
2. What are the trigonometric identities involving \( \cos \) and \( \sin \)?
3. How do you calculate sine and cosine for angles that are multiples of 18° or 36°?
4. What are other ways to express trigonometric functions in exact form?
5. How is the sine function related to the golden ratio?

**Tip:** Remember that certain angles, like 18° and 36°, have exact trigonometric values tied to the golden ratio.

Learn how to evaluate the cosine of 2/5π with a detailed step-by-step trigonometric solution. This problem involves using trigonometric identities and formulas to find the exact value of the cosine function at a non-standard angle. Suitable for advanced high school students.

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