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Can you explain what the number after the 2nd equal sign is? i know the 2nd number is a rationalized version of the degree, but then whats the number after that and how did they get that?

The image shows the trigonometric values of \( \sin 45^\circ \), \( \cos 45^\circ \), and \( \tan 45^\circ \) in degrees and radians, along with their rationalized forms.

Let's break down the third number in the equations for \( \sin 45^\circ \) and \( \cos 45^\circ \), which is \( \frac{\sqrt{2}}{2} \):

1. **Step 1: Understanding the exact value**  
   The angles are converted into radians:
   - \( \sin 45^\circ = \sin \frac{\pi}{4} \)
   - \( \cos 45^\circ = \cos \frac{\pi}{4} \)
   
   Both of these trigonometric values equal \( \frac{1}{\sqrt{2}} \), which is their non-rationalized form.

2. **Step 2: Rationalizing the denominator**  
   The expression \( \frac{1}{\sqrt{2}} \) involves a square root in the denominator. To eliminate the square root from the denominator (rationalizing it), you multiply both the numerator and denominator by \( \sqrt{2} \):
   \[
   \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
   \]
   
   This is the third number you're asking about—\( \frac{\sqrt{2}}{2} \). It is the rationalized form of \( \frac{1}{\sqrt{2}} \), and it is used because it avoids square roots in the denominator, which is considered more standard and simplified in most mathematical contexts.

### Summary of the Steps:
- \( \frac{1}{\sqrt{2}} \) is the trigonometric value of \( \sin 45^\circ \) and \( \cos 45^\circ \).
- To rationalize the denominator, you multiply the fraction by \( \frac{\sqrt{2}}{\sqrt{2}} \), giving \( \frac{\sqrt{2}}{2} \).

Would you like more details or further clarification on this? 

Here are 5 relative questions:
1. What is rationalizing a denominator and why is it important in mathematics?
2. Can all trigonometric ratios be simplified similarly for other common angles?
3. How do we convert from degrees to radians in trigonometry?
4. What are the trigonometric values for \( \sin 30^\circ \) and \( \cos 30^\circ \), and how are they derived?
5. Why do we often use radians instead of degrees in calculus and higher mathematics?

**Tip:** In trigonometry, memorizing the values for common angles like 30°, 45°, and 60° can greatly speed up calculations!

Can you explain what the number after the 2nd equal sign is? I know the 2nd number is a rationalized version of the degree, but then what's the number after that and how did they get that?

Learn how to rationalize trigonometric values for sin(45°), cos(45°), and tan(45°) by converting from radians to simplified forms like √2/2. This problem focuses on rationalizing the denominator in trigonometric expressions, suitable for high school math students.

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