Find unknown angles from measured trig ratios. Also compute sin, cos, tan from any angle. Quick inputs, clear solutions, and easy sharing tools today.
Trigonometric ratios relate an angle θ to a right triangle: sin(θ)=opposite/hypotenuse, cos(θ)=adjacent/hypotenuse, tan(θ)=opposite/adjacent.
To recover an angle from a ratio, inverse functions are used: θ=arcsin(x), θ=arccos(x), θ=arctan(x). Multiple angles can share the same ratio due to periodicity.
| Angle (deg) | sin(θ) | cos(θ) | tan(θ) | Notes |
|---|---|---|---|---|
| 30 | 0.500000 | 0.866025 | 0.577350 | Common special-angle values. |
| 45 | 0.707107 | 0.707107 | 1.000000 | Diagonal of a square. |
| 60 | 0.866025 | 0.500000 | 1.732051 | Equilateral triangle split in half. |
| 90 | 1.000000 | 0.000000 | undefined | tan(90°) is undefined because cos(90°)=0. |
Tip: For exact angles, keep more decimals in inputs.
Sine, cosine, and tangent connect an angle to a shape. In a right triangle they compare side lengths, and on the unit circle they map an angle to a point. These ratios power everything from roof pitch and ramps to rotating machinery, waves, and signals. They also describe periodic motion in vibration and acoustics.
Degrees are convenient for everyday geometry, while radians simplify many formulas in calculus and physics. One full turn is 360° or 2π radians, so conversion is straightforward. If a formula uses angular frequency, phase, or derivatives, radians are usually the safer choice. When results look “off by a factor,” a unit mismatch is often the reason.
On the unit circle, the x‑coordinate equals cos(θ) and the y‑coordinate equals sin(θ). That single picture explains signs in each quadrant and why values repeat every 360° (or 2π). It also helps you predict results before calculating, which is useful for spotting typos. Reference angles help you estimate values without memorizing every case.
Tangent is sin(θ)/cos(θ). Whenever cosine is zero, tangent cannot be computed, such as at 90° and 270°. A good calculator should warn you instead of printing a misleading large number. If you see an “undefined” result, it usually indicates a vertical slope. Near these angles, tiny errors cause big changes.
To recover an angle, you can use arcsin, arccos, or arctan, but each has a restricted principal range. For full‑quadrant angles, arctan2(y, x) is preferred because it uses both coordinates. If you start from a ratio, consider whether multiple angles could share the same value. Also check domain rules: arcsin and arccos require inputs between −1 and 1.
For a ladder, sin(θ) relates height to length, while cos(θ) relates base distance to length. For bearings and navigation, cosine and sine project a distance into east and north components. In electronics, sine waves describe AC voltages, and cosine often appears as a phase‑shifted form.
The most common error is mixing degrees and radians. Another is forgetting signs in different quadrants, especially after simplifying a negative angle. Quick checks help: sin(0)=0, cos(0)=1, tan(45°)=1, and values must stay between −1 and 1 for sine and cosine. If results violate these bounds, recheck inputs.