Key takeaways
- SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- sin θ, cos θ, and tan θ describe a right triangle's shape from one angle alone.
- tan θ = sin θ ÷ cos θ, and it's undefined wherever cos θ = 0 (at 90° and 270°).
- sin(45°) and cos(45°) are equal, both about 0.7071.
In this guide
What are sine, cosine, and tangent?
In a right triangle, pick one of the non-right angles, θ (theta). Sine, cosine, and tangent compare that angle's opposite, adjacent, and hypotenuse sides, remembered with SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why do sine, cosine, and tangent matter?
These three ratios let you find heights and distances you can't measure directly. Once you know an angle, you know the shape of the triangle it belongs to — no tape measure required for the other sides.
At exactly 45°, the opposite and adjacent sides of a right triangle are the same length, which is why sin(45°) and cos(45°) come out identical, both about 0.7071.
The formula
| Symbol | What it means | Example |
|---|---|---|
| θ (theta) | The angle you're finding ratios for, in degrees | 45° |
| sin θ | Opposite side ÷ hypotenuse | 0.7071 |
| cos θ | Adjacent side ÷ hypotenuse | 0.7071 |
| tan θ | Opposite side ÷ adjacent side (also sin θ ÷ cos θ) | 1 |
How to find sin, cos, and tan step by step
- 1Convert θ to radians if your tool needs it. See the degrees to radians calculator.
- 2Calculate sin θ.
- 3Calculate cos θ.
- 4Calculate tan θ by dividing sin θ by cos θ (undefined when cos θ = 0, at 90° and 270°).
Enter your own angle below to see all three ratios calculated step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Mixing up which side is opposite vs adjacent. Opposite is across from the angle; adjacent is next to it (but isn't the hypotenuse).
- Assuming tan is always defined. At 90°, cos θ = 0, so tan θ is undefined (it shoots off toward infinity).
- Reading off the wrong ratio. Double-check which of the three values (sin, cos, or tan) a problem is actually asking for.
Tips and tricks
- If you know a ratio and want the angle back, use the inverse trig calculator.
- For right-triangle side lengths without needing any angle at all, see the Pythagorean theorem calculator.
- Sanity-check your result: sin θ and cos θ should always be between -1 and 1.
Where you'll use it in real life
- Construction: finding a building's height from its shadow length and the sun's angle.
- Navigation: breaking a travel direction into forward and sideways components.
- Engineering: analyzing forces acting at an angle, like tension in a cable.
- Video games and animation: rotating and positioning objects on screen.
Quick summary
Sine, cosine, and tangent are three ratios built from a right triangle's sides and one angle, remembered with SOHCAHTOA. They let you find missing distances from an angle alone. Try the calculator above with your own angle to see all three values.
Worked example
A ladder leans against a wall, making a 45° angle with the ground. Before you measure any actual lengths, what are the sine, cosine, and tangent of that 45° angle?
- 01θ = 45° = 0.7854 rad
- 02sin 45° = 0.7071
- 03cos 45° = 0.7071
- 04tan 45° = 1
- 05Result shown = sin 45° = 0.7071
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does SOHCAHTOA mean?
It's a memory aid: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent — the three basic trig ratios for a right triangle.
Why is sin and cos of 45° the same number?
At 45°, the opposite and adjacent sides of a right triangle are equal in length, so both ratios to the hypotenuse come out the same: about 0.7071.
When is tan undefined?
Tangent is undefined at 90° and 270°, where cosine equals zero, since tan θ = sin θ ÷ cos θ and you can't divide by zero.
Do I need a right triangle to use sin, cos, and tan?
The basic SOHCAHTOA definitions come from right triangles, but sine, cosine, and tangent are defined for any angle using the unit circle, including angles beyond 90°.
