$$ \sin\frac15\pi $$$$  $$の値を求めよ

Given the expression $ \sin\left(\frac{1}{5}\pi\right) $, we are asked to find its exact value.

### Step-by-Step Solution:

1. **Convert the Angle to Degrees:**
   \[
   \frac{1}{5} \pi \text{ radians} = \frac{1}{5} \times 180^\circ = 36^\circ
   \]

2. **Use Known Values for Sine of Standard Angles:**
   The sine of 36° is a well-known value in trigonometry. It can be derived using the half-angle identity or from the pentagon’s geometry. The exact value is:
   \[
   \sin(36^\circ) = \frac{\sqrt{10 - 2\sqrt{5}}}{4}
   \]

Thus, the exact value of $ \sin\left(\frac{1}{5}\pi\right) $ is:

\[
\sin\left(\frac{1}{5}\pi\right) = \frac{\sqrt{10 - 2\sqrt{5}}}{4}
\]

Would you like more details or have any questions?

---

Here are some related questions:
1. How can you derive the exact value of $ \sin(36^\circ) $?
2. What are the exact trigonometric values for angles related to a regular pentagon?
3. How can you use the half-angle formula to find trigonometric values?
4. What is the relationship between sine and cosine for complementary angles?
5. How is the sine function related to other angles like 54° or 18°?

**Tip:** Understanding the geometric properties of polygons like pentagons can help in deriving exact trigonometric values for angles like 36°.

Explore how to find the exact value of sin(1/5π) using trigonometric identities and known values. This problem involves converting radians to degrees and using trigonometric identities for non-standard angles. Ideal for advanced high school students and those interested in trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation