The STUDYQUERIES unit circle calculator is a very useful online tool that automatically calculates radians, sine values, cosine values, and tangent values based on the angle of the circle. Essentially, a unit circle or trigonometry circle is a circle with a radius of one unit.

How to Use the Estimate the Unit Circle Calculator?

It is easy and quick to use the unit circle calculator. Just follow the steps below to use the tool.

  • Step 1: Enter the Angle of the Unit Circle (in degrees) in the first input box.
  • Step 2: Click on “Solve”
  • Step 3: Check the “Radians”, “Sine Function Value”, “Cos Function Value”, and “Tan Function Value” for the entered angle in the output boxes.

For an angle \(45\ degrees\), the calculator will give the output as:

\(Radians = 0.785\)

\(\sin 45°= 0.707\)

\(\cos 45° = 6.123233995736766 \times 10^{-17}\)

\(\tan 45° = 1.000\)

Unit Circle Calculator

What is Unit Circle?

A unit circle is a circle with a unit radius from its name alone. A circle is a closed geometric figure that has no sides or angles. In addition to having all the properties of a circle, the unit circle is also derived from the equation of a circle. In addition, a unit circle is useful for calculating the standard angles of all trigonometric ratios.

Here we shall learn the equation of the unit circle, and understand how to represent each of the points on the circumference of the unit circle, with the help of trigonometric ratios of \(\cos \theta\) and \(\sin \theta\).

Binary Multiplication – Rules & Examples

One unit circle is a circle with a radius of one unit. A unit circle is generally represented in cartesian coordinates. The unit circle can be represented algebraically using the second-degree equation with two variables, x, and y. In trigonometry, the unit circle is useful for finding the trigonometric ratios sine, cosine, and tangent.

Unit Circle Definition

The locus of a point which is at a distance of one unit from a fixed point is called a unit circle.

Equation of a Unit Circle

The general equation of a circle is \((x – a)^2 + (y – b)^2 = r^2\), which represents a circle having the center \((a, b)\) and the radius \(r\). This equation of a circle is simplified to represent the equation of a unit circle. A unit circle is formed with its center at the point \((0, 0)\), which is the origin of the coordinate axes. and a radius of 1 unit. Hence the equation of the unit circle is \((x – 0)^2 + (y – 0)^2 = 1^2\). This is simplified to obtain the equation of a unit circle.

Equation of a Unit Circle
Equation of a Unit Circle

Equation of a Unit Circle: \(x^2 + y^2 = 1\)

Here for the unit circle, the center lies at \((0,0)\) and the radius is \(1\ unit\). The above equation satisfies all the points lying on the circle across the four quadrants.

Finding Trigonometric Functions Using a Unit Circle

Using a unit circle, we can calculate the trigonometric functions sine, cosine, and tangent. Applying Pythagoras’s theorem in a unit circle will help us understand trigonometric functions. Imagine a right triangle placed in a unit circle in the cartesian coordinate system.

Finding Trigonometric Functions Using a Unit Circle
Finding Trigonometric Functions Using a Unit Circle

The radius of the circle represents the hypotenuse of the right triangle. The radius vector makes an angle \(\theta\) with the positive x-axis and the coordinates of the endpoint of the radius vector are \((x, y)\). Here the values of \(x\) and \(y\) are the lengths of the base and the altitude of the right triangle. Now we have a right angle triangle with the sides \(1, x, y\). Applying this in trigonometry, we can find the values of the trigonometric ratio, as follows:

\(\sin \theta = \frac{Altitude}{Hypoteuse} = \frac{y}{1}\)
\(\cos \theta = \frac{Base}{Hypotenuse} = \frac{x}{1}\)

We now have \(\sin \theta = y,\  \cos \theta = x\), and using this we now have \(\tan \theta = \frac{y}{x}\). Similarly, we can obtain the values of the other trigonometric ratios using the right-angled triangle within the unit circle. Also by changing the \(\theta\) values we can obtain the principal values of these trigonometric ratios.

Unit Circle with Sin Cos and Tan

Any point on the unit circle has coordinates \((x, y)\), which are equal to the trigonometric identities of \((\cos \theta, \sin \theta)\). For any values of \(\theta\) made by the radius line with the positive x-axis, the coordinates of the endpoint of the radius represent the cosine and the sine of the \(\theta\) values. Here we have \(\cos \theta = x,\ and\ \sin \theta = y\), and these values are helpful to compute the other trigonometric ratio values. Applying this further we have \(tanθ = \frac{\sin \theta}{\cos \theta}\ or\  \tan \theta = \frac{y}{x}\).

Another important point to be understood is that the \(sin \theta\ and\  \cos \theta\) values always lie between \(1\ and\ -1\), and the radius value is \(1\), and it has a value of \(-1\) on the negative x-axis. The entire circle represents a complete angle of \(360º\) and the four-quadrant lines of the circle make angles of \(90º, 180º, 270º, 360º(0º)\). At \(90º\) and at \(270º\) the \(\cos \theta\) value is equal to \(0\) and hence the tan values at these angles are undefined.

\(\mathbf{\color{red}{Find\ the\ value\ of\ tan\ 45º\ using\ sin\ and\ cos\ values\ from\ the\ unit\ circle.}}\)

We know that \(\tan 45° = \frac{\sin 45°}{\cos 45°}\)

Using the unit circle chart:

\(\sin 45° = \frac{1}{\sqrt{2}}\)
\(\cos 45° = \frac{1}{\sqrt{2}}\)

Therefore, \(\tan 45° = \frac{\sin 45°}{\cos 45°}\)

\(= \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}\)

\(= 1\)

Therefore, \(\tan 45° = 1\)

Unit Circle Chart in Radians

The unit circle represents a complete angle of \(2\pi\) radians. And the unit circle is divided into four quadrants at angles of \(\frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ and\ 2\pi\) respectively. Further within the first quadrant at the angles of \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\) are the standard values, which are applicable to the trigonometric ratios. The points on the unit circle for these angles represent the standard angle values of the cosine and sine ratios.

Unit Circle Chart In Radians
Unit Circle Chart In Radians

A closer look at the figure above reveals that the values are repeated across the four quadrants but with a change of sign. This change in sign is caused by the reference x-axis and y-axis, which are positive on one side and negative on the other. With this, we can easily find the trigonometric ratio values of standard angles in each of the four quadrants of the unit circle.

Unit Circle and Trigonometric Identities

The unit circle identities of sine, cosecant, and tangent can be further used to obtain the other trigonometric identities such as cotangent, secant, and cosecant. The unit circle identities such as cosecant, secant, cotangent are the respective reciprocal of the sine, cosine, tangent. Further, we can obtain the value of \(\tan \theta\) by dividing \(\sin \theta\) with \(\cos \theta\), and we can obtain the value of \(\cot \theta\) by dividing \(\cos \theta\) with \(sin \theta\).

For a right triangle placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring \(1, x, y\) units respectively, the unit circle identities can be given as,

\(\sin \theta = \frac{y}{1}\)

\(cos \theta = \frac{x}{1}\)

\(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}\)

\(\sec \theta = \frac{1}{x}\)

\(\csc \theta = \frac{1}{y}\)

\(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{x}{y}\)

Unit Circle Pythagorean Identities

It is easy to understand and prove the three Pythagorean identities of trigonometric ratios using the unit circle. Pythagoras’s theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Below are the three Pythagorean identities in trigonometry.

$$\sin^2(\theta) + \cos^2(\theta) = 1$$
$$1 + \tan^2(\theta) = \sec^2(\theta)$$
$$1 + \cot^2(\theta) = \csc^2(\theta)$$

Here we shall try to prove the first identity with the help of the Pythagoras theorem. Let us take \(x\) and \(y\) as the legs of the right-angled triangle having a hypotenuse \(1\ unit\).

Applying Pythagoras theorem we have \(x^2 + y^2 = 1\) which represents the equation of a unit circle. Also in a unit circle, we have, \(x = \cos \theta\), and \(y = \sin \theta\), and applying this in the above statement of the Pythagoras theorem, we have \(\cos^2(\theta)  + \sin^2(\theta) = 1\). Thus we have successfully proved the first identity using the Pythagoras theorem. Further within the unit circle, we can also prove the other two Pythagorean identities.

Unit Circle and Trigonometric Values

The various trigonometric identities and their principal angle values can be calculated through the use of a unit circle. In the unit circle, we have cosine as the x-coordinate and sine as the y-coordinate. Let us now find their respective values for \(\theta = 0°\), and \(\theta = 90º\).

Principal Values Using Unit Circle
Principal Values Using Unit Circle

For \(\theta = 0°\), the x-coordinate is \(1\) and the, y-coordinate is \(0\). Therefore, we have \(\cos 0º = 1\), and \(\sin 0º = 0\). Let us look at another angle of \(90º\). Here the value of \(\cos 90º = 1\), and \(\sin 90º = 1\).

Further, let us use this unit circle and find the important trigonometric function values of \(\theta\) such as \(30º, 45º, 60º\). Also, we can also measure these \(\theta\) values in radians. We know that \(360° = 2\pi\ radians\). We can now convert the angular measures to radian measures and express them in terms of the radians.

Unit Circle Table

The unit circle table is used to list the coordinates of the points on the unit circle that correspond to common angles with the help of trigonometric ratios.

Unit Circle Table
Unit Circle Table

We can find the secant, cosecant, and cotangent functions also using these formulas:

$$\sec \theta = \frac{1}{\cos \theta}$$
$$\csc \theta = \frac{1}{\sin \theta}$$
$$\cot \theta = \frac{1}{\tan \theta}$$

We have discussed the unit circle for the first quadrant. Similarly, we can extend and find the radians for all the unit circle quadrants. The numbers \(1/2, 1/√2, √3/2, 0, 1\) repeat along with the sign in all 4 quadrants.

Unit Circle in Complex Plane

A unit circle consists of all complex numbers of absolute value as \(1\). Therefore, it has the equation of \(|z| = 1\). Any complex number \(z = x +iy\) will lie on the unit circle with the equation given as \(x^2 + y^2 = 1\).

The unit circle can be considered as unit complex numbers in a complex plane, i.e., the set of complex numbers z given by the form,

$$z = e^{it}=cos t +i(sin t) = cis(t)$$

The relation given above represents Euler’s formula.


Where is negative pi on the unit circle?

The interval (−π\2,π\2) is the right half of the unit circle. Negative angles rotate clockwise, so this means that −π\2 would rotate π\2 clockwise, ending up on the lower y-axis (or as you said, where 3π\2 is located).

How do you use the unit circle?

A unit circle can be used to define right triangle relationships known as sine, cosine, and tangent. These relationships describe how angles and sides of a right triangle relate to one another. Say, for example, we have a right triangle with a 30-degree angle, and whose longest side, or hypotenuse, is a length of 7.

What is the formula for a unit circle?

The unit circle is the circle of radius 1 that is centered at the origin. The equation of the unit circle is x²+y²=1.

What is the positive and negative angle?

An angle is a measure of rotation. Angles are measured in degrees. Positive angles result from counterclockwise rotation, and negative angles result from clockwise rotation. An angle with its initial side on the x‐axis is said to be in standard position.

What is the unit circle used for in real life?

It can be used to calculate distances like the heights of mountains or how far away the stars in the sky are. The cyclic, repeated nature of trig functions means that they are useful for studying different types of waves in nature: not just in the ocean, but the behavior of light, sound, and electricity as well.