The six trigonometric values are based on the unit circle. For an angle \(\theta\), the point
on the unit circle is:
\[
(x,y)=(\cos\theta,\sin\theta)
\]
This means cosine is the \(x\)-coordinate and sine is the \(y\)-coordinate.
1. Main trigonometric functions
\[
\sin\theta=y
\]
\[
\cos\theta=x
\]
\[
\tan\theta=\frac{\sin\theta}{\cos\theta}
\]
Tangent is undefined when \(\cos\theta=0\), because division by zero is not allowed.
2. Reciprocal trigonometric functions
The other three trigonometric functions are reciprocal functions:
\[
\csc\theta=\frac{1}{\sin\theta}
\]
\[
\sec\theta=\frac{1}{\cos\theta}
\]
\[
\cot\theta=\frac{\cos\theta}{\sin\theta}
\]
These are undefined whenever their denominator is zero.
3. Exact values for standard angles
Standard angles such as \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\), and their
coterminal versions have exact values. For example:
\[
225^\circ=\frac{5\pi}{4}
\]
It is in Quadrant III with reference angle \(45^\circ\). Therefore:
\[
\sin225^\circ=-\frac{\sqrt2}{2}
\]
\[
\cos225^\circ=-\frac{\sqrt2}{2}
\]
\[
\tan225^\circ=1
\]
4. Quadrant signs
| 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. Formula summary
| Function |
Formula |
Undefined when |
| Sine |
\(\sin\theta=y\) |
Never undefined on the unit circle |
| Cosine |
\(\cos\theta=x\) |
Never undefined on the unit circle |
| Tangent |
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) |
\(\cos\theta=0\) |
| Cosecant |
\(\csc\theta=\dfrac{1}{\sin\theta}\) |
\(\sin\theta=0\) |
| Secant |
\(\sec\theta=\dfrac{1}{\cos\theta}\) |
\(\cos\theta=0\) |
| Cotangent |
\(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\) |
\(\sin\theta=0\) |
6. Calculator mode for non-standard angles
If an angle is not a standard unit-circle angle, exact radical values are usually not simple.
In that case, a decimal approximation is used. For example, \(37^\circ\) has decimal sine,
cosine, and tangent values.
7. Common mistakes
- Do not switch sine and cosine: cosine is \(x\), sine is \(y\).
- Remember that tangent is \(\sin\theta/\cos\theta\).
- Check the quadrant before assigning signs.
- Do not give a value for tangent when \(\cos\theta=0\).
- Do not give a value for cotangent or cosecant when \(\sin\theta=0\).
- Use radians or degrees consistently when entering the angle.
