Trigonometry calculator
Give one acute angle and the side adjacent to it, and the rest of the right triangle follows.
Result
- Opposite side
- 5.7735
- Hypotenuse
- 11.5470
- Other acute angle
- 60.00°
- sin
- 0.500000
- cos
- 0.866025
- tan
- 0.577350
Worked examples
Finding the far side
A 30° angle with an adjacent side of 10.
- tan 30° = 0.5774
- opposite = 10 × 0.5774 = 5.7735
- hypotenuse = 10 ÷ cos 30° = 11.547
Opposite 5.77, hypotenuse 11.55.
The three ratios
In a right triangle, each acute angle fixes the ratios between the sides. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — the mnemonic SOH-CAH-TOA encodes exactly this.
"Adjacent" and "opposite" are defined relative to the angle you are working with, not to the triangle. Swap to the other acute angle and the two sides swap roles, which is why sine and cosine of complementary angles are equal.
Degrees and radians
Degrees divide a full turn into 360 parts; radians measure the arc length on a unit circle, so a full turn is 2π. Most calculators can work in either, and mixing them is the most common cause of an answer that is wildly wrong rather than slightly wrong.
To convert, multiply degrees by π/180. A 30° angle is π/6 radians, roughly 0.5236.
Formula
tan θ = opposite ÷ adjacent • cos θ = adjacent ÷ hypotenuse
Exact values worth memorising
These five angles cover most textbook work and are exact rather than rounded.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.5 | √3/2 ≈ 0.8660 | 1/√3 ≈ 0.5774 |
| 45° | √2/2 ≈ 0.7071 | √2/2 ≈ 0.7071 | 1 |
| 60° | √3/2 ≈ 0.8660 | 1/2 = 0.5 | √3 ≈ 1.7321 |
| 90° | 1 | 0 | undefined |
Choosing the right ratio
Pick the ratio that contains the side you know and the side you want.
| You know | You want | Use | Rearranged |
|---|---|---|---|
| Angle + adjacent | Opposite | tan | opposite = adjacent × tan θ |
| Angle + adjacent | Hypotenuse | cos | hypotenuse = adjacent ÷ cos θ |
| Angle + opposite | Adjacent | tan | adjacent = opposite ÷ tan θ |
| Angle + opposite | Hypotenuse | sin | hypotenuse = opposite ÷ sin θ |
| Angle + hypotenuse | Opposite | sin | opposite = hypotenuse × sin θ |
| Two sides | The angle | inverse ratio | θ = arctan(opposite ÷ adjacent) |
Degrees and radians
| Degrees | Radians | Decimal |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 360° | 2π | 6.2832 |
Frequently asked questions
What is SOH-CAH-TOA?
A mnemonic for the three ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
How do I find a side if I know an angle and one other side?
Pick the ratio containing both the side you know and the one you want. Knowing the adjacent side and wanting the opposite, use tangent: opposite = adjacent × tan θ.
Why is tan 90° undefined?
Tangent is opposite divided by adjacent, and at 90° the adjacent side has length zero. Division by zero has no value, so the tangent is undefined rather than infinite.
Do these ratios work for triangles without a right angle?
Not directly. For other triangles use the law of sines or the law of cosines, which generalise the same relationships.
How do I find an angle from two sides?
Use the inverse function. If you know opposite and adjacent, the angle is arctan(opposite ÷ adjacent).
What is the difference between sin and arcsin?
Sine takes an angle and returns a ratio; arcsine takes a ratio and returns an angle. They undo each other.
Why is my calculator giving strange trig answers?
Almost always a degrees-versus-radians mismatch. sin(30) is 0.5 in degrees but −0.988 in radians.
Do these ratios work for non-right triangles?
Not directly. Use the law of sines or the law of cosines, which extend the same relationships to any triangle.
What does SOH-CAH-TOA stand for?
Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
How do I convert degrees to radians?
Multiply by π/180. 30° is π/6, about 0.5236 radians.