Trigonometry Calculator
Trigonometry relates the angles of a triangle to the lengths of its sides. This calculator solves for missing sides and angles in both right-angled and general triangles.
Formula
sin θ = opposite ÷ hypotenuse · cos θ = adjacent ÷ hypotenuse · tan θ = opposite ÷ adjacent
- Remembered as SOH CAH TOA
- Check whether the calculator is in degrees or radians before blaming the working
Opposite 3, hypotenuse 5: sin θ = 0.6, so θ = 36.87°
Right-Angled Triangles: SOH CAH TOA
- sinθ = Opposite ÷ Hypotenuse
- cosθ = Adjacent ÷ Hypotenuse
- tanθ = Opposite ÷ Adjacent
In a 3-4-5 triangle, the angle opposite the side of 3 has sinθ = 0.6, so θ = 36.9°.
The hypotenuse is always the longest side, opposite the right angle. Opposite and adjacent are defined relative to the angle you are working with, which is why they swap when you switch angles.
Triangles Without a Right Angle
SOH CAH TOA only works with a right angle present. For any other triangle:
Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C
Cosine rule: c² = a² + b² − 2ab cos C
Use the sine rule when you have a matching side and opposite angle. Use the cosine rule when you know two sides and the angle between them, or all three sides.
Degrees, Radians and the Ambiguous Case
The commonest error is a calculator in the wrong mode. Degrees divide a circle into 360; radians use 2π. Higher mathematics prefers radians, school geometry uses degrees, and mixing them silently produces wrong answers.
Note also the ambiguous case: when using the sine rule to find an angle, two valid answers may exist, since sin 30° and sin 150° are equal. Check which fits the triangle you actually have.
Quick Reference
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 0.5 | 0.8660 | 0.5774 |
| 45° | π/4 | 0.7071 | 0.7071 | 1 |
| 60° | π/3 | 0.8660 | 0.5 | 1.7321 |
| 90° | π/2 | 1 | 0 | undefined |
Frequently Asked Questions
- What is SOH CAH TOA?
- A mnemonic for the three ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
- When do I use the cosine rule instead of the sine rule?
- Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule when you have a side and its opposite angle.
- Why does my calculator give the wrong answer?
- Almost always because it is in radian mode when the question is in degrees, or the reverse. Check the mode before trusting any result.
- What is the ambiguous case?
- When finding an angle with the sine rule, two answers can be valid because sine gives the same value for an angle and its supplement, such as 30 and 150 degrees.