Trigonometry Calculator
Calculate sin, cos and tan of any angle in degrees, radians or gradians — or enter a ratio and find the angle using arcsin, arccos or arctan.
Find sin, cos and tan of an Angle — or the Angle from a Ratio
Trigonometry Formula
Where:
- \(\theta\) — the angle, which can be given in degrees, radians or gradians
- \(y\) — the known ratio, i.e. the value of \(\sin\theta\), \(\cos\theta\) or \(\tan\theta\)
- \(\arcsin\), \(\arccos\), \(\arctan\) — the inverse trigonometric functions, also written \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\)
The first line defines the three basic trigonometric functions as ratios of the sides of a right triangle. The second line inverts the process: given a ratio, the inverse functions recover the angle.
Trigonometry – Calculation Example
Example: find the sine, cosine and tangent of a 30° angle (the calculator's default values).
First convert the angle to radians, since the trigonometric functions are defined on the unit circle:
\[ 30^\circ \cdot \frac{\pi}{180^\circ} = \frac{\pi}{6} \approx 0.523599 \text{ rad} \]
\[ \sin 30^\circ = 0.5 \qquad \cos 30^\circ \approx 0.866025 \qquad \tan 30^\circ = \frac{0.5}{0.866025} \approx 0.577350 \]
Result: \(\sin\theta = 0.5\), \(\cos\theta \approx 0.866025\), \(\tan\theta \approx 0.577350\).
Inverse example: if you know that \(\sin\theta = 0.5\), switch the mode to "Ratio (sin, cos or tan) → angle", choose sin θ and enter 0.5. The calculator returns the principal angle \(\theta = \arcsin(0.5) = 30^\circ \approx 0.523599\) rad.
How to Use the Calculator
- In the Calculation mode dropdown, choose Angle → sin, cos, tan if you know the angle, or Ratio (sin, cos or tan) → angle if you know one of the ratios.
- For the forward mode: enter the angle in the Angle field and pick its unit (deg, rad or grad) from the Unit dropdown. Any real number is accepted — the angle does not need to lie between 0° and 360°.
- For the inverse mode: select which ratio you know in Known function (sin θ, cos θ or tan θ) and enter its value in Ratio value.
- Read the results: the calculator shows \(\sin\theta\), \(\cos\theta\), \(\tan\theta\) and the angle expressed in degrees, radians and gradians.
What Are Sine, Cosine and Tangent?
Trigonometry connects angles to ratios of lengths. In a right triangle, the sine of an acute angle is the length of the side opposite the angle divided by the hypotenuse; the cosine is the adjacent side divided by the hypotenuse; and the tangent is the opposite side divided by the adjacent side. These definitions extend beyond right triangles through the unit circle: if a point on a circle of radius 1 makes an angle \(\theta\) with the positive x-axis, then its coordinates are \((\cos\theta, \sin\theta)\).
This circular definition works for any angle — negative, larger than 360°, or expressed in radians or gradians. The tangent has a simpler identity worth remembering: \(\tan\theta = \sin\theta / \cos\theta\). It grows without bound as \(\cos\theta\) approaches zero, which is why the tangent is undefined at 90°, 270°, and generally at \(90^\circ + k \cdot 180^\circ\).
The inverse functions run the logic backwards. If you know, for example, that \(\sin\theta = 0.5\), then \(\theta = \arcsin(0.5) = 30^\circ\). This is essential in geometry (finding an angle from side lengths), physics (projectile motion, waves) and engineering (signal analysis, AC circuits). The root mean square of a sinusoidal signal, for instance, relies on values of \(\sin\theta\) and \(\cos\theta\).
Trigonometric functions also appear far outside geometry. Complex numbers are written in polar form as \(r(\cos\theta + i\sin\theta)\), which is why tools like the complex conjugate and complex root calculators involve angles and radii in their computations.
Common Values Reference Table
| θ (degrees) | θ (radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
Assumptions and Limitations
- Principal values only. Every ratio corresponds to infinitely many angles. In inverse mode the calculator returns the principal value: \(\arcsin\) and \(\arctan\) give results in −90°…90°, while \(\arccos\) gives results in 0°…180°. Other solutions follow from the identities \(\sin\theta = \sin(180^\circ - \theta)\) and \(\cos\theta = \cos(360^\circ - \theta)\), plus adding any multiple of 360°.
- Range of sine and cosine. Since \(\sin\theta\) and \(\cos\theta\) always lie between −1 and 1, the calculator rejects ratio values outside this interval for these two functions. The tangent has no such restriction — any real ratio is valid.
- Undefined tangent. When \(\cos\theta = 0\) (at 90° + k·180°), the tangent does not exist and the calculator reports it as undefined rather than returning an enormous number.
- Rounding. Results are rounded to 6 decimal places, and values that differ from zero only by floating-point noise (for example \(\sin 180^\circ\) computed as 10⁻¹⁶) are displayed as exactly 0.
- Unit awareness. Always check the Unit dropdown before entering an angle. Entering 3.14 while the unit is set to degrees will give a very different answer than 3.14 radians. Recall that \(180^\circ = \pi\) radians = 200 gradians.
Practical Tips
When solving triangle problems, decide first whether you know an angle or a ratio. If two sides of a right triangle are known, form the appropriate ratio (opposite/hypotenuse, adjacent/hypotenuse or opposite/adjacent) and use inverse mode to recover the angle. If the angle is known and you need a side, forward mode gives the ratio, which you then multiply by the hypotenuse or adjacent side.
For angles outside 0°…360°, remember that all three functions are periodic: sine and cosine repeat every 360° (2π rad), and the tangent repeats every 180° (π rad). The calculator handles such inputs directly, so a 750° angle or a −45° angle produces correct values without any manual reduction.
Frequently Asked Questions
How do I calculate sin, cos and tan by hand without a calculator?
Memorize the key angles: 0°, 30°, 45°, 60°, 90°. At 45°, \ and \. For 30° and 60°, remember \, \ and swap them for 60°. For other angles, use the unit circle with reference angles, or trigonometric identities such as \ to break an angle into known parts. You can also interpolate between known table values for quick estimates.
What is the difference between degrees, radians and gradians?
A full circle is 360 degrees, \ radians, or 400 gradians. One radian is the angle where the arc length equals the radius, so \ radians. Gradians divide a right angle into 100 parts, so \. Radians dominate calculus and physics because derivatives of trig functions take their simplest form in that unit. Always confirm the Unit dropdown matches your input before reading the result.
Why is tangent undefined at 90°?
Because \ and at \ the cosine equals zero, making the division impossible — the tangent shoots toward infinity on one side and negative infinity on the other. The same happens at \ and generally every \. The calculator reports this as undefined rather than a huge number, since floating-point noise can make the tangent look enormous but meaningless.
Why does the inverse mode return only one angle when many are possible?
Every ratio matches infinitely many angles because the functions are periodic. To keep results unambiguous, inverse functions return only the principal value: \ and \ give angles in \, while \ gives \. If \, both 30° and 150° work, plus any of these plus \. Use the identities \ and \ to find the rest.
What are the most common mistakes when using trigonometry?
The biggest one is a unit mismatch: entering \ while the calculator is set to degrees instead of radians gives a completely wrong answer. Others include forgetting that sine and cosine must stay within −1 to 1 (a ratio like 2.5 has no real angle), confusing the opposite and adjacent sides when building a ratio, and assuming \ means \ — it denotes the inverse function (arcsin), not the reciprocal, which is cosecant.
How do I find an angle from two sides of a right triangle?
Use the SOH-CAH-TOA rule to build a ratio, then apply the matching inverse function. If the side opposite the unknown angle is 5 and the hypotenuse is 10, then \, so \. Opposite over adjacent gives the tangent, adjacent over hypotenuse gives the cosine. Enter the ratio in the calculator's inverse mode with the correct function selected to get the angle immediately.
What do negative values of sin, cos or tan mean?
They indicate the angle's direction on the unit circle. Sine is negative below the x-axis (angles between 180° and 360°), cosine is negative left of the y-axis (between 90° and 270°), and tangent is negative wherever sine and cosine have opposite signs. For example, \ because a negative angle rotates clockwise. Negative angles and angles above 360° are handled directly — the functions are periodic, so no manual reduction is needed.