Sine, cosine & tangent
In a right triangle, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent (remembered as SOH CAH TOA). On the unit circle, cos θ is the x-coordinate, sin θ is the y-coordinate, and tan θ = sin θ / cos θ.
| Use it to | Formula |
|---|---|
| Sine (SOH) | sin θ = opposite / hypotenuse |
| Cosine (CAH) | cos θ = adjacent / hypotenuse |
| Tangent (TOA) | tan θ = opposite / adjacent = sin θ / cos θ |
| Identity | sin² θ + cos² θ = 1 |
Sine, cosine & tangent calculator
Three ratios. One right triangle. Change a value and the diagram, result, and working update together.
Make an angle. See the connection.
Quadrant I · both coordinates are positive.
One circle. Three trigonometric functions.
The unit circle has a radius of 1. Starting at the positive x-axis, an angle θ moves counterclockwise around the circle. The point you land on is (cos θ, sin θ). Tangent is the ratio sin θ / cos θ, so it is undefined where cosine is zero.
Use the number field or arrow keys on the slider for precise, keyboard-accessible exploration. Special angles show exact values alongside decimal approximations.
The idea
Cosine is adjacent divided by hypotenuse. Tangent is opposite divided by adjacent. On the unit circle the hypotenuse is 1, so cosine becomes the x-coordinate and sine becomes the y-coordinate. Tangent equals sine divided by cosine.
Worked examples
1Find a side with sine
- A ramp is 10 m long and rises at 30°.
- height = 10 × sin 30° = 10 × 0.5.
Answer: The ramp rises 5 m.
2Find an angle with tangent
- Opposite = 3 and adjacent = 4.
- tan θ = 3 / 4 = 0.75, so θ = arctan(0.75).
Answer: θ ≈ 36.87°.
Common mistakes
- Opposite and adjacent depend on which acute angle you choose. The hypotenuse is always the side opposite the right angle. For angles beyond 90°, the unit circle extends these definitions using signed coordinates.
Make the formula move.
Open the full tool for a larger workspace.