Enter any two side lengths of a right triangle (opposite, adjacent, and/or hypotenuse), relative to angle a, into the Trig Ratio Calculator. The calculator will evaluate the missing side and display the sin(a), cos(a), and tan(a) ratios.
Trigonometric Ratios for a Right Triangle
For angle a, the side labels are always relative to that angle: the opposite side is across from the angle, the adjacent side touches the angle, and the hypotenuse is the longest side opposite the right angle. Enter any two side lengths in the same unit to find the third side and the three primary trig ratios.
Core Formulas
\sin(a)=\frac{O}{H},\quad \cos(a)=\frac{A}{H},\quad \tan(a)=\frac{O}{A}| Ratio | Mnemonic | What it compares | Best use |
|---|---|---|---|
| Sine | SOH | Opposite to hypotenuse | When you know or need the side across from angle a |
| Cosine | CAH | Adjacent to hypotenuse | When you know or need the side next to angle a |
| Tangent | TOA | Opposite to adjacent | When comparing vertical change to horizontal change |
Finding the Missing Side
| Known sides | Formula | Notes |
|---|---|---|
| Opposite and adjacent | H=\sqrt{O^2+A^2} |
Use this when both legs are known and you need the hypotenuse. |
| Opposite and hypotenuse | A=\sqrt{H^2-O^2} |
Use this when the side across from angle a and the hypotenuse are known. |
| Adjacent and hypotenuse | O=\sqrt{H^2-A^2} |
Use this when the side next to angle a and the hypotenuse are known. |
Finding the Angle from a Ratio
a=\sin^{-1}\left(\frac{O}{H}\right),\quad a=\cos^{-1}\left(\frac{A}{H}\right),\quad a=\tan^{-1}\left(\frac{O}{A}\right)If you already know the side ratio, inverse trig functions return angle a.
Useful Checks
H^2=O^2+A^2,\quad \tan(a)=\frac{\sin(a)}{\cos(a)}| Check | What it means |
|---|---|
| Hypotenuse is largest | The hypotenuse must be longer than both the opposite and adjacent sides. |
| Same input unit | All side lengths should use the same unit before calculating ratios. |
| Ratios are unitless | Because one length is divided by another, the units cancel out. |
| Side labels depend on angle a | Changing the reference angle changes which side is opposite and adjacent. |
Quick Reference Values
The values below assume the listed opposite and adjacent sides are measured relative to angle a.
| Opposite (O) | Adjacent (A) | Hypotenuse (H) | sin(a) | cos(a) | tan(a) |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 0.6000 | 0.8000 | 0.7500 |
| 5 | 12 | 13 | 0.3846 | 0.9231 | 0.4167 |
| 8 | 15 | 17 | 0.4706 | 0.8824 | 0.5333 |
| 1 | 1 | 1.4142 | 0.7071 | 0.7071 | 1.0000 |
Common Mistakes
- Using side labels from the wrong reference angle.
- Entering side lengths in mixed units without converting first.
- Choosing a value for the hypotenuse that is not the longest side.
- Confusing tangent with sine or cosine; tangent compares only the two legs.
