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 California Common Core State Standards. It is aligned to Standard F-TF.1, F-TF.2, which requires students to understand radian measure and use the unit circle to extend the trigonometric functions.
Radian measure defines an angle by arc length over radius, so a full turn is \(2\pi\); on the unit circle an angle \(\theta\) lands on the point \((\cos\theta,\sin\theta)\).
Theory
A radian is the angle at the center of a circle that subtends an arc equal in length to the radius. Because the full circumference is \(2\pi r\), a full turn is
The unit circle is the circle of radius 1 centered at the origin. For an angle \(\theta\) in standard position, the terminal point is
which is how the trigonometric functions are defined for any angle, not just those in a right triangle.
Conversions and arc length:
How to convert and use radians
- Degrees \(\to\) radians: multiply by \(\dfrac{\pi}{180^\circ}\) and simplify.
- Radians \(\to\) degrees: multiply by \(\dfrac{180^\circ}{\pi}\).
- Unit-circle point: read \((\cos\theta,\sin\theta)\).
- Arc length: apply \(s=r\theta\) with \(\theta\) in radians.
Multiply by \(\dfrac{\pi}{180^\circ}\).
| \(60^\circ\cdot\dfrac{\pi}{180^\circ}\) | \(=\) | \(\dfrac{60\pi}{180}\) |
| \(=\) | \(\dfrac{\pi}{3}\) |
Multiply by \(\dfrac{180^\circ}{\pi}\).
| \(\dfrac{3\pi}{4}\cdot\dfrac{180^\circ}{\pi}\) | \(=\) | \(\dfrac{3\cdot 180^\circ}{4}\) |
| \(=\) | \(135^\circ\) |
Use \(s=r\theta\) with \(\theta\) in radians.
| \(s\) | \(=\) | \(12\cdot\dfrac{\pi}{6}\) |
| \(=\) | \(2\pi\) |
The point is \((\cos\theta,\sin\theta)\).
| \((\cos\dfrac{\pi}{2},\ \sin\dfrac{\pi}{2})\) | \(=\) | \((0,\ 1)\) |
Common pitfalls
Frequently asked questions
What is a radian?
The angle whose arc length equals the radius. A full circle is \(2\pi\) radians, which equals \(360^\circ\).
How do you convert between degrees and radians?
Degrees to radians: multiply by \(\pi/180\). Radians to degrees: multiply by \(180/\pi\).
What is the unit circle?
The circle of radius 1 centered at the origin. An angle \(\theta\) in standard position lands on the point \((\cos\theta,\sin\theta)\).
What is the arc length formula?
\(s=r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.