The Simple Finger Trick for Trigonometry Values
Do you struggle to remember the values of sine, cosine, and tangent for common angles like 0°, 30°, 45°, 60°, and 90°? Here's a clever and easy finger trick that can help you recall these fundamental trigonometric values quickly!
Understanding the Finger Representation
Imagine your left hand with your palm facing you and your fingers extended. Each finger represents a specific angle, starting from the thumb:
- Thumb: 0°
- Index Finger: 30°
- Middle Finger: 45°
- Ring Finger: 60°
- Pinky Finger: 90°
Finding the Sine (sin) Value
To find the sine of an angle using this trick:
- Fold down the finger corresponding to the angle you want to find the sine of.
- Count the number of fingers to the left of the folded finger.
- The sine of the angle is the square root of this number, divided by 2.
Formula:
$\sin \theta = \frac{\sqrt{\text{number of fingers to the left}}}{2}$
Example for $\sin 30^\circ$:
- Fold the index finger (30°).
- There is 1 finger to the left (the thumb).
- $\sin 30^\circ = \frac{\sqrt{1}}{2} = \frac{1}{2}$
Finding the Cosine (cos) Value
To find the cosine of an angle using this trick:
- Fold down the finger corresponding to the angle you want to find the cosine of.
- Count the number of fingers to the right of the folded finger.
- The cosine of the angle is the square root of this number, divided by 2.
Formula:
$\cos \theta = \frac{\sqrt{\text{number of fingers to the right}}}{2}$
Example for $\cos 30^\circ$:
- Fold the index finger (30°).
- There are 3 fingers to the right (middle, ring, pinky).
- $\cos 30^\circ = \frac{\sqrt{3}}{2}$
Finding the Tangent (tan) Value
The finger trick doesn't directly give you the tangent value. However, you can easily calculate it using the sine and cosine values you found:
Formula:
$\tan \theta = \frac{\sin \theta}{\cos \theta}$
Example for $\tan 30^\circ$:
- We know $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$.
- $\tan 30^\circ = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$
Important Note: This finger trick is a fantastic way to quickly remember the sine and cosine values for these common angles. Remember the relationship $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to find the tangent values.
Summary Table
| Angle ($\theta$) | Folded Finger | Fingers to the Left (for sin) | $\sin \theta$ | Fingers to the Right (for cos) | $\cos \theta$ | $\tan \theta = \frac{\sin \theta}{\cos \theta}$ |
|---|---|---|---|---|---|---|
| 0° | Thumb | 0 | $\frac{\sqrt{0}}{2} = 0$ | 4 | $\frac{\sqrt{4}}{2} = 1$ | $\frac{0}{1} = 0$ |
| 30° | Index Finger | 1 | $\frac{\sqrt{1}}{2} = \frac{1}{2}$ | 3 | $\frac{\sqrt{3}}{2}$ | $\frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$ |
| 45° | Middle Finger | 2 | $\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$ | 2 | $\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$ | $\frac{1/\sqrt{2}}{1/\sqrt{2}} = 1$ |
| 60° | Ring Finger | 3 | $\frac{\sqrt{3}}{2}$ | 1 | $\frac{\sqrt{1}}{2} = \frac{1}{2}$ | $\frac{\sqrt{3}/2}{1/2} = \sqrt{3}$ |
| 90° | Pinky Finger | 4 | $\frac{\sqrt{4}}{2} = 1$ | 0 | $\frac{\sqrt{0}}{2} = 0$ | $\frac{1}{0}$ (undefined) |
This handy finger trick can be a great aid in quickly recalling these essential trigonometric values. Practice it a few times, and you'll have them at your fingertips!