To get \(\frac{1}{\sqrt{2}}\), one common situation involves working with trigonometric functions, particularly when dealing with angles in a right triangle or in the unit circle. Here's how you might arrive at this value:

### Scenario: Unit Circle or Trigonometric Functions
In the unit circle, the sine and cosine of a 45-degree angle (or \(\frac{\pi}{4}\) radians) are both \(\frac{1}{\sqrt{2}}\). Here's why:

1. **Isosceles Right Triangle**: Consider a right triangle where both legs are of equal length (an isosceles right triangle). If each leg has a length of \(1\), the hypotenuse will be \(\sqrt{1^2 + 1^2} = \sqrt{2}\).

2. **Trigonometric Ratios**: The sine and cosine of the 45-degree angle (or \(\frac{\pi}{4}\) radians) in this triangle are given by:
   \[
   \sin(45^\circ) = \cos(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
   \]

### Simplification
In some cases, \(\frac{1}{\sqrt{2}}\) might be further simplified by rationalizing the denominator:
\[
\frac{1}{\sqrt{2}} = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}
\]

This value \(\frac{1}{\sqrt{2}}\) is therefore a key number in trigonometry, particularly in the context of angles like 45° in the unit circle.

Would you like more details on this, or do you have any further questions?

### Related Questions:
1. What are some real-world applications of the value \(\frac{1}{\sqrt{2}}\)?
2. How does \(\frac{1}{\sqrt{2}}\) relate to the unit circle and trigonometric identities?
3. What are the steps to rationalize the denominator in expressions like \(\frac{1}{\sqrt{2}}\)?
4. How can the value \(\frac{1}{\sqrt{2}}\) be used in vector mathematics?
5. What are the key differences between using \(\frac{1}{\sqrt{2}}\) and \(\frac{\sqrt{2}}{2}\)?

### Tip:
In trigonometry, angles like 45°, 30°, and 60° often lead to common fractional values like \(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{3}}{2}\). These are important to recognize for simplifying trigonometric expressions.

Explore how to derive and simplify \(\frac{1}{\sqrt{2}}\) using trigonometric functions and the unit circle. This explanation covers the properties of a 45-degree angle in a right triangle and its application in trigonometric identities. Suitable for high school students.

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