Unit Circle Calculator

The online unit circle calculator allows you to determine the sine, cosine, and tangent value for an angle that helps to figure out the coordinates on the unit circle. When it comes to circle angle calculations, it is important to have an exact idea about the appropriate unit circle values.

No doubt, remembering sine, cosines, or unit circle tangent values seems like a daunting task. But, this unit circle point calculator helps you to find the circle trigonometry functions corresponding to the unit circle chart.

Also, use a free online circumference calculator to figure out the circumference, diameter, radius, area, sphere surface area & sphere volume of a circle.

What is a Unit Circle?

A unit circle is a circle with 1 radius. In other words, the center of a unit circle is at \((0,0)\) and its radius is 1. Total length of its intercepted arc is equivalent to the radian measure of the central \(angle t\).if \((x,y)\) are the endpoint on the unit circle of any arc that has length s. Then \((x,y)\) coordinates of that point can be labelled as functions of the angle as seen in the image below:

(image)

Unit Circle Equation:

The equation for the unit circle is: \(u2 + v2 = 1\)

Unit Circle in Radians & Degrees:

For a unit circle encountered angles measured in terms of radians and degrees. A unit circle chart shows the position of all the points along the unit circle that are made when we divide the circle into eight and twelve parts. It demonstrates all the radians and circles. With the support of terminal point calculator, it becomes easy to find all these angels and degrees.

Unit Circle Chart:

(chart)

Unit Circle Tangent, Sine, & Cosine:

Unit circle tangent values can be remembered only by memorizing the definition of the tangent.

(image)

Right triangle: It is the exact illustration of the tangent definition. Contradictory side over an adjacent. It is the ratio of all the opposite and adjacent sides to an angle in a right-angled triangle.
By observing above image: \(tan(α) = opposite/adjacent\).
Tangent can also be figured out when its sine divided by its cosine: \(tan(α) = sin(α)/cos(α) = y/x\).

Unit circle relations for sine and cos:

(image)

While examining unit circle, Sine is the y-coordinate whereas x-coordinate is cosine.

Let’s assume any point A on the circumference of the unit circle. At this point the coordinates will be \(x\) and \(y\). Its radius will be 1 as it is a unit circle. Now if you project the radius on its \(x\) and \(y axes\), you will get an exact right triangle. Wherever \(|x|\) and \(|y|\) are the lengths, and its hypotenuse will beequal to 1.

Like every right triangle, values of the trigonometric functions can be calculated by finding the side ratios:\(sin(α) = opposite/hypotenuse = y/1 = y\).

image

Therefore, we can conclude that sine is the \(y-coordinate\). And in the same way \(cos(α) = adjacent/hypotenuse =x/1 = cosine is the x-coordinate\).

image

Hence the equation of the unit circle, defined by the Pythagorean theorem will be: \(x² + y² = 1\).
It can also be represented as: \(sin²(α) + cos²(α) = 1. T\)
find tangent sine andcosine of any unit circle the best way it to use a unit circle calculator so that all the possible risks and errors can be avoided.

How to Memorize the Unit Circle?

When it comes to a unit circle there are two things that needs to be remembered. The first one is Angle conversion. It will assist you in understanding that how to change between an angle in degrees and in terms of radians. The second is trigonometric functions of the popular angles. A unit circle chart might help you a lot in this regard. Some popular angels are:

\(30° = π/6\)
\(45° = π/4\)
\(60° = π/3\)
\(90° = π/2\)
\(180° = π\)
\(360° = 2π\)

In case of some other angle, there is formula for angle conversion: \(radians = π/180° x\) degrees. With these angels and formula conversion of the unit circle’s radians to degrees will be easy for anyone.
Following equations might be helpful for memorizing unit circle:

\(tanθ=cosθ/sinθ\)
\(cotθ=sinθ/cos\),
\(secθ=1/cosθ1\),
\(cscθ=1/sinθ1\)

Furthermore, the following displays commonly used angles:

(chart)

To avoid the hustle of remembering all these you can use unit circle calculator. it will be a great help to understand and memorize the unit circle values.

How to Find Trig Functions with Unit Circle Calculator?

Using the calculator for the unit circle is stress-free and give fast results by following these steps:

Input:

In the first step you have to Enter the Angle of the Unit Circle.
Now select “\(deg\)” “\(rad\)” or “\(pie rad\)” from the drop-down menu according to the requirement.
Click the calculate button.

Output:

The calculator will display values of:

\(Sin\)
\(Cos\)
\(tan\)

FAQ’s:

What is the unit circle used for in real life?

There are many applications of the unit circle in front of the normal eye. For example, whenever you buy a pizza you don’t pay attention that how they cut it. The person who was cutting the pizza has to use unit circle to cut it right.

How do you solve a unit circle problem?

To solve any problem that involves a unit circle you can have three ways:

Use of a unit circle calculator.
Use of unit circle chart.
Apply unit circle formula.

Who invented the unit circle?

The first recognized table of chords was formed by the Greek mathematician Hipparchus in about \(140 BC\). Though these tables have not persisted, but it is claimed that there were total 12 books of tables of chords were written by Hipparchus. These bools laid the foundations of unit circle and unit circle chart.

How do special right triangles create the unit circle?

All the measures of the inner angles of a right triangle add up to \(180º\). One angles from all of these has a measure that is equal to \(90º\). But the other two angles of a right triangle must be acute angles. For doing this, you must implant a right triangle into a circle.

Why we use Sin Cos Tan?

Sin Cos and Tan are fundamentally just functions that share an angle with a ratio of two sides in any right triangle. Sin is equal to the side that is opposite to the angle that you are conducting the functions on over the hypotenuse which is in fact the lengthiest side in the triangle.

End-Note:

This unit circle calculator aids you to find out the coordinates of any point on the unit circle. All you have to do is to enter the angel and chose the degree. It will display sine and cosine values of that angel. This unit circle solver takes the responsibility of deliver the accurate results without any cost. This is one of the best option for academics and learning purposes. While using unit circle calculator students and professors don’t have to memorize any values from unit circle chart.

References:

A source of Wikipedia: All you need to know about the unit circle

From the source of khanacademy: Unit: Trigonometric functions, intro to radians & much more!

From the source of clarku: tangent to the circle