Introduction
We are used to measuring angles in degrees, where a full circle is . This number comes from ancient Babylonian astronomy (approximating days in a year). In mathematics, we prefer a measure based on the circle's own geometry: the Radian.
Prerequisite Connection
Recall that the circumference of a circle is . The "angle" for a full circle corresponds to this arc length.
Today's Increment
We define radian as the angle formed when the arc length equals the radius. A full circle is radians.
Why This Matters
Calculus does not work in degrees. The derivative formulas for sine and cosine (e.g., ) are only true if is measured in radians. If you use degrees, messy constants appear.
Key Concepts
Conversion Factors
Since , we use this ratio to convert units, just like converting feet to inches.
- Degrees to Radians: Multiply by .
- Radians to Degrees: Multiply by .
Worked Examples
Example 1: Degrees to Radians
Convert to radians.
Set up the multiplication
We need to cancel the "degrees" unit, so degrees goes on the bottom.
Simplify the fraction
Example 2: Radians to Degrees
Convert radians to degrees.
Set up multiplication
We need to cancel the , so goes on bottom.
Simplify
Cancel the terms. Calculate .
Example 3: Degrees without explicit multiples
Convert radians to degrees (decimal).
Identify the unit
Wait, where is the ? It's not required! 2 radians is just the number 2.
Apply Formula
Multiply by .
Common Pitfalls
Confusing the fraction direction
Students often flip and . Remember: you want the "old" unit to be on the bottom so it cancels out.
Real-World Application
Computer Graphics & Gaming
Game engines like Unity and Unreal Engine process all rotations in radians (or quaternions which rely on them). If you tell a character to rotate "90", the engine might think you mean 90 radians (which is spinning around 14 times!), not 90 degrees.
Practice Quiz
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