\#3.) Why is it valid to use the conversion factor $$ \frac{180}{\pi} $$ when converting radians to degrees?

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To understand why it is valid to use the conversion factor $$ \frac{180}{\pi} $$ when converting radians to degrees, we need to look at the relationship between radians and degrees.

1. **Definition of Radians and Degrees**:
   - A full circle is defined as $$ 360 $$ degrees.
   - A full circle is also defined as $$ 2\pi $$ radians.

2. **Establishing the Conversion Factor**:
   - Since $$ 2\pi $$ radians correspond to $$ 360 $$ degrees, we can set up the following relationship:
     $$
     2\pi \text{ radians} = 360 \text{ degrees}
     $$

3. **Finding the Conversion Factor**:
   - To find the conversion factor from radians to degrees, we can divide both sides of the equation by $$ 2\pi $$:
     $$
     1 \text{ radian} = \frac{360}{2\pi} \text{ degrees}
     $$
   - Simplifying $$ \frac{360}{2\pi} $$ gives:
     $$
     1 \text{ radian} = \frac{180}{\pi} \text{ degrees}
     $$

4. **Using the Conversion Factor**:
   - Therefore, to convert an angle from radians to degrees, you multiply the angle in radians by $$ \frac{180}{\pi} $$:
     $$
     \text{Degrees} = \text{Radians} \times \frac{180}{\pi}
     $$

This derivation shows that the conversion factor $$ \frac{180}{\pi} $$ is valid because it is derived from the fundamental relationship between radians and degrees based on the measurement of a full circle.
To convert radians to degrees, use the factor $$ \frac{180}{\pi} $$ because a full circle is $$ 2\pi $$ radians and $$ 360 $$ degrees. So, $$ 1 $$ radian equals $$ \frac{180}{\pi} $$ degrees.

\#3.) Why is it valid to use the conversion factor \( \frac{180}{\pi} \) when converting radians to degrees?

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