Algebra 2
Trigonometric functions (introduction)
Radian measure and the unit circle
20 practice questions
0 video lessons
Theory + worked examples
Theory
A radian measures an angle as arc length divided by radius, so a half turn is \(\pi\):
\[180^\circ=\pi\text{ radians},\qquad s=r\theta.\]
On the unit circle (radius 1), an angle \(\theta\) meets the circle at \((\cos\theta,\sin\theta)\).
Convert with \(\dfrac{\pi}{180^\circ}\) or \(\dfrac{180^\circ}{\pi}\).
\(\theta\) lands on \((\cos\theta,\sin\theta)\).
Radians and arc length.
Conversions and arc length:
\[\text{rad}=\deg\cdot\dfrac{\pi}{180^\circ},\qquad s=r\theta\]
A full circle is \(2\pi\) radians.
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).
Example 1 — Degrees to radians
Convert \(60^\circ\) to radians.
Solution
Multiply by \(\dfrac{\pi}{180^\circ}\).
| \(60^\circ\cdot\dfrac{\pi}{180^\circ}\) | \(=\) | \(\dfrac{\pi}{3}\) |
Example 2 — Radians to degrees
Convert \(\dfrac{\pi}{6}\) to degrees.
Solution
Multiply by \(\dfrac{180^\circ}{\pi}\).
| \(\dfrac{\pi}{6}\cdot\dfrac{180^\circ}{\pi}\) | \(=\) | \(30^\circ\) |
Example 3 — A point on the circle
Find the coordinates at \(\theta=\dfrac{\pi}{2}\).
Solution
Use \((\cos\theta,\sin\theta)\).
| \(\left(\cos\dfrac{\pi}{2},\sin\dfrac{\pi}{2}\right)\) | \(=\) | \((0,1)\) |
Example 4 — Arc length
Find the arc length for \(\theta=\dfrac{\pi}{3}\) on a circle of radius \(6\).
Solution
Use \(s=r\theta\).
| \(s\) | \(=\) | \(6\cdot\dfrac{\pi}{3}\) |
| \(=\) | \(2\pi\) |
Common pitfalls
Arc length \(s=r\theta\) needs radians, not degrees.
\(\pi\) radians is \(180^\circ\), not \(360^\circ\).
On the unit circle, \(x=\cos\theta\) and \(y=\sin\theta\).
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.
More in Trigonometric functions (introduction)