Math101 · Independent Ontario learning library
Math101learn.math101.caUnit Circle
The unit circle turns every angle into a point whose coordinates are cosine and sine.
On the unit circle, the point reached by an angle $\theta$ is $(\cos\theta,\sin\theta)$.
One circle organizes trigonometry
The unit circle has centre $(0,0)$ and radius $1$:
Begin at $(1,0)$ and rotate counterclockwise through $\theta$. The endpoint is
Cosine is horizontal; sine is vertical.
The triangle connection
Drop a vertical line from the circle point to the $x$-axis. The right triangle has hypotenuse $1$, horizontal leg $x$, and vertical leg $y$:
The Pythagorean Theorem gives
Degrees and radians
One full turn is $360^\circ=2\pi$ radians. Therefore
Radians are natural on a unit circle: an angle of $\theta$ radians cuts off an arc of length $\theta$.
First-quadrant anchor angles
| Degrees | Radians | $\cos\theta$ | $\sin\theta$ |
|---|---|---|---|
| $0^\circ$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $\pi/6$ | $\sqrt3/2$ | $1/2$ |
| $45^\circ$ | $\pi/4$ | $\sqrt2/2$ | $\sqrt2/2$ |
| $60^\circ$ | $\pi/3$ | $1/2$ | $\sqrt3/2$ |
| $90^\circ$ | $\pi/2$ | $0$ | $1$ |
Sine rises from $0$ to $1$; cosine uses the same values in reverse.
Signs by quadrant
| Quadrant | Coordinate signs | Positive functions |
|---|---|---|
| I | $(+,+)$ | sine and cosine |
| II | $(-,+)$ | sine |
| III | $(-,-)$ | neither sine nor cosine |
| IV | $(+,-)$ | cosine |
Tangent is $\sin\theta/\cos\theta$, so it is positive when the coordinates share a sign.
Reference angles
A reference angle is the acute angle between the terminal side and the $x$-axis. It gives coordinate magnitudes; the quadrant supplies signs.
Coterminal angles
Angles ending at the same point differ by whole turns:
in radians, or $\theta+360^\circ k$ in degrees, where $k$ is an integer. Thus $30^\circ$, $390^\circ$, and $-330^\circ$ are coterminal.
Radian example
Common mistakes
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is sin(30°)? Give an exact value.
- The point at 30° is (√3/2, 1/2).
- Sine is the y-coordinate.
- sin(30°) = 1/2.
What is cos(π)?
- The angle π lands at the point (−1, 0).
- Cosine is the x-coordinate.
- cos(π) = −1.
End of lesson
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