Radian measure and the unit circle
Radian Measure and the Unit Circle
Radian Measure and the Unit Circle is the opening topic of Trigonometric Functions in the Common Core State Standards. It is aligned to Standard F-TF.1, which requires students to understand radian measure as arc length on the unit circle and use it to extend the trigonometric functions.
A radian measures an angle by arc length over radius, so \(180^\circ=\pi\); on the unit circle an angle \(\theta\) lands on \((\cos\theta,\sin\theta)\).
Theory
A radian measures an angle as arc length divided by radius, so a half turn is \(\pi\):
On the unit circle (radius 1), an angle \(\theta\) meets the circle at \((\cos\theta,\sin\theta)\).
Conversions and arc length:
How to work in radians
- To convert degrees to radians, multiply by \(\dfrac{\pi}{180^\circ}\).
- To convert radians to degrees, multiply by \(\dfrac{180^\circ}{\pi}\).
- Read circle coordinates as \((\cos\theta,\sin\theta)\).
- Use \(s=r\theta\) for arc length (with \(\theta\) in radians).
Multiply by \(\dfrac{\pi}{180^\circ}\).
| \(60^\circ\cdot\dfrac{\pi}{180^\circ}\) | \(=\) | \(\dfrac{\pi}{3}\) |
Multiply by \(\dfrac{180^\circ}{\pi}\).
| \(\dfrac{\pi}{6}\cdot\dfrac{180^\circ}{\pi}\) | \(=\) | \(30^\circ\) |
Use \((\cos\theta,\sin\theta)\).
| \(\left(\cos\dfrac{\pi}{2},\sin\dfrac{\pi}{2}\right)\) | \(=\) | \((0,1)\) |
Use \(s=r\theta\).
| \(s\) | \(=\) | \(6\cdot\dfrac{\pi}{3}\) |
| \(=\) | \(2\pi\) |
Common pitfalls
Frequently asked questions
What is a radian?
An angle measure equal to arc length divided by radius.
How many radians is \(180^\circ\)?
\(\pi\) radians.
What point does an angle reach on the unit circle?
\((\cos\theta,\sin\theta)\).
What is the arc length formula?
\(s=r\theta\), with \(\theta\) in radians.