Trigonometry Calculator

Calculate trigonometric values in degrees, radians, or gradians with unit circle and graph previews.

Results are calculated automatically as you enter data.

Use numbers or expressions such as pi/4, 2*pi, or 45+15.

Angle or expression

Function Angle unit

Decimal places

Formula notes

Angle conversionDegrees and gradians are converted to radians before evaluating direct functions.

Inverse functionsasin, acos, and atan return an angle in the selected unit.

Result -

Function graphDirect functions over 0 to 4pi

Unit circleCurrent angle position

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Assumptions and limitations

Results are educational estimates, not professional engineering, construction, safety, scientific, legal, financial, or academic advice. No guarantee is made that outputs are accurate, complete, suitable, or error-free. Verify units, formulas, assumptions, tolerances, rounding, and limitations independently, and use qualified professional review for technical or safety-sensitive work. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Are Trigonometric and Inverse Trigonometric Functions?

Trigonometric functions connect an angle to ratios and coordinates. On the unit circle, an angle $\theta$ points to a location on a circle with radius $1$. The coordinates of that point are:

$$ (\cos \theta,\ \sin \theta) $$

That simple idea explains why sine and cosine are so useful. The cosine tells you the horizontal coordinate, and the sine tells you the vertical coordinate. From those two values, the other common trigonometric functions can be built.

Inverse trigonometric functions go in the opposite direction. Instead of starting with an angle and finding a ratio, they start with a ratio and return an angle. For example, $\sin(30^\circ)=0.5$, so $\arcsin(0.5)=30^\circ$ when the answer is expressed in degrees.

A trigonometry calculator is useful because many angles do not have simple exact values. It also helps convert between degrees, radians, and gradians, which is one of the most common sources of mistakes in trigonometry.

Why Trigonometry Matters

Trigonometry appears anywhere angles, circular motion, waves, slopes, or repeating patterns are involved. Students commonly meet it in geometry, algebra, precalculus, calculus, physics, and engineering contexts.

The most important practical benefit is that trigonometry turns angle information into measurable quantities. If you know an angle and one side of a triangle, you can often find another side. If you know a point on a unit circle, you can find sine and cosine values. If you know a ratio, an inverse trigonometric function can help recover the angle that produced it.

Key Terms to Know

Angle: A measure of rotation. In this calculator, direct trigonometric functions can use degrees, radians, or gradians.
Degree: An angle unit where a full turn is $360^\circ$.
Radian: An angle unit based on arc length. A full turn is $2\pi$ radians.
Gradian: An angle unit where a full turn is $400$ gradians, so a right angle is $100$ gradians.
Unit circle: A circle centered at the origin with radius $1$. It connects angles to the coordinates $(\cos \theta,\sin \theta)$.
Sine and cosine: Functions that give the vertical and horizontal coordinates on the unit circle.
Tangent: The ratio of sine to cosine, written $\tan \theta$.
Cotangent, secant, and cosecant: Reciprocal trigonometric functions.
Inverse trigonometric function: A function such as $\arcsin$, $\arccos$, or $\arctan$ that returns an angle from a ratio.
Asymptote: A line that a graph approaches but does not cross. Tangent, cotangent, secant, and cosecant have undefined values at certain angles because of vertical asymptotes.

How Trigonometric Evaluation Works

For direct trigonometric functions, the angle is first understood in the selected unit. Radians are the natural input unit for most trigonometric formulas, so degrees and gradians are converted to radians before evaluation.

Degrees to radians:

$$ \theta_{\text{rad}}=\theta_{\text{deg}}\times \frac{\pi}{180} $$

Gradians to radians:

$$ \theta_{\text{rad}}=\theta_{\text{grad}}\times \frac{\pi}{200} $$

Once the angle is in radians, sine and cosine can be understood from the unit circle:

$$ (\cos \theta,\sin \theta) $$

The other direct functions are defined from sine and cosine:

$$ \tan \theta=\frac{\sin \theta}{\cos \theta} $$

$$ \cot \theta=\frac{\cos \theta}{\sin \theta} $$

$$ \sec \theta=\frac{1}{\cos \theta} $$

$$ \csc \theta=\frac{1}{\sin \theta} $$

These formulas also explain why some results are undefined. If a denominator is $0$, the function cannot return a finite real value. For example, $\tan(90^\circ)$ is undefined because $\cos(90^\circ)=0$.

Inverse trigonometric functions reverse part of this relationship. They answer questions such as “What angle has this sine value?” or “What angle has this tangent value?” The three inverse functions supported here are $\arcsin$, $\arccos$, and $\arctan$.

For $\arcsin$ and $\arccos$, the input must be between $-1$ and $1$ because sine and cosine never produce values outside that range:

$$ -1 \le x \le 1 $$

The inverse tangent function accepts any real number because tangent can produce any real value on its principal branch.

Examples of Trigonometric Evaluation in Practice

Example 1: Evaluating a Sine Value in Degrees

Suppose you want to evaluate:

$$ \sin(45^\circ) $$

First convert the angle to radians:

$$ 45^\circ \times \frac{\pi}{180}=\frac{\pi}{4} $$

Then evaluate the sine:

$$ \sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2} $$

As a decimal, this is approximately:

$$ 0.70710678 $$

This result is a dimensionless ratio, not a length or an angle.

Example 2: Using an Inverse Function

Suppose the input is $0.5$ and the selected function is $\arcsin$.

The question is: what angle has a sine value of $0.5$?

$$ \arcsin(0.5)=\frac{\pi}{6} $$

In degrees:

$$ \frac{\pi}{6}\times \frac{180}{\pi}=30^\circ $$

In gradians:

$$ \frac{\pi}{6}\times \frac{200}{\pi}=\frac{100}{3}\approx 33.333333\text{ grad} $$

The same mathematical angle can be displayed in different units.

Example 3: An Undefined Tangent Value

A right angle can be written as $90^\circ$, $\frac{\pi}{2}$ radians, or $100$ gradians.

For tangent:

$$ \tan \theta=\frac{\sin \theta}{\cos \theta} $$

At a right angle:

$$ \sin\left(\frac{\pi}{2}\right)=1 $$

$$ \cos\left(\frac{\pi}{2}\right)=0 $$

So:

$$ \tan\left(\frac{\pi}{2}\right)=\frac{1}{0} $$

Division by zero is undefined, so tangent does not have a finite value there. The same issue occurs for secant whenever cosine is zero, and for cotangent and cosecant whenever sine is zero.

Example 4: Reading Unit-Circle Coordinates

If $\theta=60^\circ$, then:

$$ 60^\circ\times \frac{\pi}{180}=\frac{\pi}{3} $$

On the unit circle:

$$ \cos\left(\frac{\pi}{3}\right)=\frac{1}{2} $$

$$ \sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2} $$

So the unit-circle point is:

$$ \left(\frac{1}{2},\frac{\sqrt{3}}{2}\right) $$

This means the horizontal coordinate is about $0.5$, and the vertical coordinate is about $0.8660254$.

How to Interpret the Result

For $\sin$, $\cos$, $\tan$, $\cot$, $\sec$, and $\csc$, the result is a dimensionless number. It represents a ratio or a value derived from the unit circle. A result such as $0.70710678$ is not “degrees” or “radians”; it is a numeric ratio.

For $\arcsin$, $\arccos$, and $\arctan$, the result is an angle. The angle is returned in the selected unit: degrees, radians, or gradians.

A valid result means the latest input, selected function, angle unit, and precision setting produced a finite value. An error message means the expression, function, unit, or mathematical domain did not allow a valid result.

Common result messages include:

Invalid input: The entered expression could not be evaluated as a finite number.
Input must be between -1 and 1: The selected function is $\arcsin$ or $\arccos$, and the input is outside the allowed domain.
Undefined value: The selected function is undefined at that input, or the result is non-finite and cannot be displayed.
Unsupported function or unit: The selected option is not one of the supported choices.

The unit-circle preview helps connect the angle to the point $(\cos \theta,\sin \theta)$. The graph preview helps show the shape of direct trigonometric functions over a repeated interval. For tangent, cotangent, secant, and cosecant, the graph preview may be visually scaled, so use it to understand the shape and asymptotes rather than exact vertical magnitude.

Common Mistakes and Misconceptions

One of the most common mistakes is using the wrong angle unit. The values $\sin(45^\circ)$ and $\sin(45\text{ rad})$ are very different because $45^\circ$ is a familiar acute angle, while $45$ radians is many full rotations plus an additional angle.

Another common mistake is entering an angle for an inverse function. The input to $\arcsin$ and $\arccos$ should be a dimensionless ratio between $-1$ and $1$, not an angle. For example, $\arcsin(30)$ is not valid in real-valued basic trigonometry because $30$ is outside the possible range of sine.

Some users also expect tangent, cotangent, secant, or cosecant to return a number at every angle. These functions involve division, so they are undefined whenever their denominator is zero.

The calculator accepts implicit multiplication, so 2pi and 2π are valid inputs. You can also use explicit multiplication, such as 2*pi or 2*π.

Rounding is another source of confusion. A displayed decimal may be rounded to the selected precision, even when the exact value is irrational. For example, $\sin(45^\circ)$ is exactly $\frac{\sqrt{2}}{2}$, but a decimal display must approximate it.

When to Use Trigonometric and Inverse Trigonometric Functions

Use direct trigonometric functions when you know an angle and need a ratio or unit-circle value. Common examples include finding the height of an object from an angle of elevation, describing circular motion, or evaluating trig expressions for homework.

Use inverse trigonometric functions when you know a ratio and need an angle. For example, if a right triangle has an opposite-to-hypotenuse ratio of $0.5$, $\arcsin(0.5)$ gives the corresponding angle on the principal branch.

Use angle conversion when the source of the problem uses one unit but the calculation or answer requires another. Degrees are common in geometry and everyday descriptions, radians are common in advanced mathematics and graphing, and gradians are useful when a problem specifically uses the $400$-gradian full-turn scale.

Limitations and Things to Keep in Mind

This calculator supports $\sin$, $\cos$, $\tan$, $\cot$, $\sec$, $\csc$, $\arcsin$, $\arccos$, and $\arctan$. It does not evaluate inverse cotangent, inverse secant, or inverse cosecant.

Only degrees, radians, and gradians are supported as angle units. Direct functions interpret the input as an angle in the selected unit. Inverse functions interpret the input as a dimensionless value and return an angle in the selected unit.

For $\arcsin$ and $\arccos$, the input must be in the interval:

$$ [-1,1] $$

Values outside that interval do not produce real-valued results for these functions.

Undefined values are expected at certain angles. Tangent and secant are undefined when cosine is $0$. Cotangent and cosecant are undefined when sine is $0$.

A finite reciprocal-trigonometric result very close to one of those singularities is shown with a warning because tiny angle-rounding changes can cause a large result change. Very large angle inputs are range-reduced, but also receive a warning when floating-point precision cannot preserve every detail of the original angle.

Precision is a display setting, not a guarantee that more decimal places create a more meaningful result. The precision setting can show from $0$ to $15$ decimal places. Very small floating-point artifacts may be shown as $0$, and values very close to integers may be displayed as integers.

The graph preview is a visual aid. Its horizontal axis is shown in radians over $0$ to $4\pi$, even if the selected input unit is degrees or gradians. Inverse functions return angle values but do not use the same periodic graph preview as direct trig functions. For tangent, cotangent, secant, and cosecant, the plotted graph may be scaled for readability.

For schoolwork, exams, engineering, or technical work, use the calculator as a checking tool and keep exact forms when required. A rounded decimal answer may not be accepted where an exact expression such as $\frac{\sqrt{3}}{2}$ or $\frac{\pi}{6}$ is expected.

How to Use This Calculator

Enter an angle or expression in the input field. Use the selected angle unit for direct functions, and use a dimensionless input for inverse functions.
Choose the function: $\sin$, $\cos$, $\tan$, $\cot$, $\sec$, $\csc$, $\arcsin$, $\arccos$, or $\arctan$.
Choose the angle unit: degrees, radians, or gradians.
Set the precision from $0$ to $15$ decimal places if you need a different display format.
Read the real-time result and status message.
Use the unit-circle preview to see the current angle vector and the coordinate values $(\cos \theta,\sin \theta)$ when the input is valid.
Use the graph preview for direct functions. When hover values are available, move near the curve to inspect approximate $x$ and $y$ values.
Use the download buttons, when present, to save the unit-circle or graph preview as a PNG image.

Frequently Asked Questions

What is the difference between $\sin$ and $\arcsin$?

The sine function starts with an angle and returns a ratio. The inverse sine function starts with a ratio and returns an angle. For example, $\sin(30^\circ)=0.5$, while $\arcsin(0.5)=30^\circ$ when degrees are selected.

Why do degrees and radians give different answers?

Degrees and radians measure angles on different scales. A full turn is $360^\circ$ but also $2\pi$ radians. If the calculator is set to radians, an input of $45$ means $45$ radians, not $45^\circ$.

Why must $\arcsin$ and $\arccos$ inputs be between $-1$ and $1$?

Sine and cosine values are coordinates on the unit circle, so they cannot be less than $-1$ or greater than $1$. Because $\arcsin$ and $\arccos$ reverse sine and cosine, their real-valued inputs are limited to that same interval.

Why is $\tan(90^\circ)$ undefined?

Tangent is defined as $\frac{\sin \theta}{\cos \theta}$. At $90^\circ$, cosine is $0$, so tangent would require division by zero. That is why the value is undefined rather than very large but finite.

What are gradians?

Gradians are an angle unit where a full turn is $400$ gradians. A right angle is $100$ gradians, so $100$ gradians equals $90^\circ$ and $\frac{\pi}{2}$ radians.

Why does the graph not always match the exact numerical scale?

For functions such as tangent, cotangent, secant, and cosecant, values can grow very large near undefined angles. A preview graph may scale these curves so their overall shape and asymptotes are easier to see. Use the numeric result for the calculated value and the graph as a visual guide.

Sources and References

Books and Textbooks

Jay Abramson et al. Precalculus 2e. OpenStax, 2021. Chapter 5, Sections 5.1–5.4, especially angle conversion, the unit circle, and the other trigonometric functions. https://openstax.org/books/precalculus-2e
Jay Abramson et al. Algebra and Trigonometry 2e. OpenStax, 2021. Chapters 7–8, especially the unit circle, reciprocal trigonometric functions, graphs of trigonometric functions, and inverse trigonometric functions. https://openstax.org/books/algebra-and-trigonometry-2e

Online and Reference Sources

National Institute of Standards and Technology. “Chapter 4: Elementary Functions,” NIST Digital Library of Mathematical Functions, Version 1.2.7, release date June 15, 2026. Used for trigonometric identities, definitions, and inverse trigonometric function reference. https://dlmf.nist.gov/4
Eric W. Weisstein. “Gradian.” MathWorld—A Wolfram Resource, last updated July 2, 2026. Used for the definition of gradians as $400$ gradians per full circle and $100$ gradians per right angle. https://mathworld.wolfram.com/Gradian.html