The Unit Circle (Foundations)
Explore the bridge between triangles and circles — the unit circle is the secret to understanding trigonometry beyond 90 degrees.
The Unit Circle is a circle with a radius of exactly 1, centered at the origin (0,0) on a coordinate plane. It allows us to calculate trig ratios for any angle, including those larger than 90°.
\[ x = \cos(\theta) \qquad y = \sin(\theta) \]Key Points on the Circle
In a circle of radius 1, the x-coordinate of any point is the cosine and the y-coordinate is the sine.
- At 0°: \((1, 0) \Rightarrow \cos(0^\circ) = 1, \sin(0^\circ) = 0\)
- At 90°: \((0, 1) \Rightarrow \cos(90^\circ) = 0, \sin(90^\circ) = 1\)
- At 180°: \((-1, 0) \Rightarrow \cos(180^\circ) = -1, \sin(180^\circ) = 0\)
- At 270°: \((0, -1) \Rightarrow \cos(270^\circ) = 0, \sin(270^\circ) = -1\)
Find the value of \(\sin(45^\circ)\).
Using the special right triangle geometry (45-45-90), we find the coordinates at 45° are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).
\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707 \]Engineers track satellites by calculating their position (x, y) along a circular path using the angle of the satellite relative to the Earth. The Unit Circle is the foundation for this spatial math.
Calculator Connection
The Unit Circle Explorer allows you to drag a point around the circle to see how the (x, y) coordinates change in real-time.
Try it with the Calculator
Apply what you've learned with this tool.