The Unit Circle and Radian Measure
Overview
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It plays a fundamental role in trigonometry because it helps define the trigonometric functions for all real numbers. In addition to the unit circle, we will also explore angle measures in both degrees and radians, which are the two primary units used in trigonometry.
Key Concepts
- Unit Circle: The unit circle is a circle with a radius of 1. Every point on the circle corresponds to an angle and gives the values of sine, cosine, and other trigonometric functions.
- Radian Measure: A radian is the angle subtended by an arc whose length is equal to the radius of the circle. There are \( 2\pi \) radians in a full circle, which corresponds to 360° in degree measure.
- Degrees and Radians Conversion: To convert from degrees to radians, multiply by \( \frac{\pi}{180} \). To convert from radians to degrees, multiply by \( \frac{180}{\pi} \).
- Trigonometric Values on the Unit Circle: The coordinates of any point on the unit circle are \( (\cos(\theta), \sin(\theta)) \), where \( \theta \) is the angle. The signs of the trigonometric ratios depend on which quadrant the angle is in.
Unit Circle Diagram
Below is a basic unit circle with commonly used angles in both radians and degrees:
- 0° = 0 radians
- 30° = \( \frac{\pi}{6} \) radians
- 45° = \( \frac{\pi}{4} \) radians
- 60° = \( \frac{\pi}{3} \) radians
- 90° = \( \frac{\pi}{2} \) radians
- 180° = \( \pi \) radians
- 270° = \( \frac{3\pi}{2} \) radians
- 360° = \( 2\pi \) radians
Practice Problems
- Convert 120° to radians.
Solution
To convert degrees to radians, use the formula:
\( 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3} \) radians.
- Convert \( \frac{3\pi}{4} \) radians to degrees.
Solution
To convert radians to degrees, use the formula:
\( \frac{3\pi}{4} \times \frac{180^\circ}{\pi} = 135^\circ \).
- Find the exact values of \( \sin(30^\circ) \) and \( \cos(30^\circ) \) using the unit circle.
Solution
From the unit circle:
\( \sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \).
- Find the exact values of \( \sin(\frac{\pi}{4}) \) and \( \cos(\frac{\pi}{4}) \) using the unit circle.
Solution
From the unit circle:
\( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \quad \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
- Find the exact values of \( \sin(\frac{\pi}{3}) \) and \( \cos(\frac{\pi}{3}) \) using the unit circle.
Solution
From the unit circle:
\( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}, \quad \cos(\frac{\pi}{3}) = \frac{1}{2} \).
- Convert \( \frac{5\pi}{6} \) radians to degrees.
Solution
To convert radians to degrees:
\( \frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ \).
- Find the exact value of \( \sin(90^\circ) \) and \( \cos(90^\circ) \).
Solution
From the unit circle:
\( \sin(90^\circ) = 1, \quad \cos(90^\circ) = 0 \).
- Convert \( 225^\circ \) to radians.
Solution
To convert degrees to radians:
\( 225^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{4} \) radians.
- Find the exact values of \( \sin(0^\circ) \) and \( \cos(0^\circ) \).
Solution
From the unit circle:
\( \sin(0^\circ) = 0, \quad \cos(0^\circ) = 1 \).
- Find the exact values of \( \sin(150^\circ) \) and \( \cos(150^\circ) \).
Solution
From the unit circle:
\( \sin(150^\circ) = \frac{1}{2}, \quad \cos(150^\circ) = -\frac{\sqrt{3}}{2} \).