The unit circle is a circle with radius \(1\) centered at the origin. It is one of the most important tools
in trigonometry because it connects angles, coordinates, and trigonometric values.
\[
x^2+y^2=1
\]
1. Unit-circle coordinates
For an angle \(\theta\), the point on the unit circle is:
\[
(x,y)=(\cos\theta,\sin\theta)
\]
This means the \(x\)-coordinate is cosine, and the \(y\)-coordinate is sine.
2. Sine, cosine, and tangent
The three main trigonometric values are:
\[
\sin\theta=y
\]
\[
\cos\theta=x
\]
\[
\tan\theta=\frac{\sin\theta}{\cos\theta}
\]
Tangent is undefined when \(\cos\theta=0\), because division by zero is not allowed.
3. Reciprocal trigonometric functions
The reciprocal functions are:
\[
\csc\theta=\frac{1}{\sin\theta}
\]
\[
\sec\theta=\frac{1}{\cos\theta}
\]
\[
\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}
\]
These values are undefined whenever their denominator is zero.
4. Quadrants and signs
The signs of sine, cosine, and tangent depend on the quadrant.
| Quadrant |
Sine |
Cosine |
Tangent |
| Quadrant I |
Positive |
Positive |
Positive |
| Quadrant II |
Positive |
Negative |
Negative |
| Quadrant III |
Negative |
Negative |
Positive |
| Quadrant IV |
Negative |
Positive |
Negative |
5. Reference angles
A reference angle is the acute angle between the terminal side and the \(x\)-axis.
It helps find exact trigonometric values in any quadrant.
For example:
\[
120^\circ
\]
is in Quadrant II, and its reference angle is:
\[
180^\circ-120^\circ=60^\circ
\]
Since \(120^\circ\) is in Quadrant II, sine is positive and cosine is negative:
\[
\sin120^\circ=\frac{\sqrt3}{2},
\qquad
\cos120^\circ=-\frac12
\]
\[
\tan120^\circ=-\sqrt3
\]
6. Common unit-circle angles
The most common unit-circle angles are built from \(30^\circ\), \(45^\circ\), and \(60^\circ\).
These correspond to:
\[
30^\circ=\frac{\pi}{6},
\qquad
45^\circ=\frac{\pi}{4},
\qquad
60^\circ=\frac{\pi}{3}
\]
Other common angles, such as \(120^\circ\), \(135^\circ\), \(150^\circ\), and \(315^\circ\),
use the same reference angles with different signs.
7. Formula summary
| Concept |
Formula |
Meaning |
| Unit circle |
\(x^2+y^2=1\) |
Circle of radius \(1\) |
| Coordinate point |
\((x,y)=(\cos\theta,\sin\theta)\) |
Cosine is \(x\), sine is \(y\) |
| Tangent |
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) |
Slope-style ratio |
| Cosecant |
\(\csc\theta=\dfrac{1}{\sin\theta}\) |
Reciprocal of sine |
| Secant |
\(\sec\theta=\dfrac{1}{\cos\theta}\) |
Reciprocal of cosine |
| Cotangent |
\(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\) |
Reciprocal of tangent |
8. Common mistakes
- Do not switch sine and cosine: cosine is the \(x\)-coordinate, sine is the \(y\)-coordinate.
- Tangent is undefined when \(\cos\theta=0\).
- Cosecant is undefined when \(\sin\theta=0\).
- Secant is undefined when \(\cos\theta=0\).
- Always check the quadrant to decide the signs of the trig values.
- Angles that differ by \(360^\circ\) or \(2\pi\) have the same unit-circle point.
