## Introduction to Rotation Calculator:

Rotation calculator geometry is an online tool that helps you to **find the rotation **geometry of a given function about its center axis. It is used to evaluate the rotating point of an object around a specific axis by a certain angle in a fraction of a second.

The rotations calculator is a helpful source for students, teachers, and researchers as it is widely used in various fields like computer graphics, robotics, engineering, and physics.

## What is Rotation?

Rotation is the **movement of an object** around a specific point or axis in mathematics. In two dimensions the rotation is present around the origin (0,0) coordinates around an axis and it is a fixed rotation of an object.

On the other hand in three-dimensional space, the object revolves around coordinate axes(x,y,z) by arbitrary angle. It is measured in degrees or radians. One rotation of an object is around with 360° or 2πradians angle.

## Formula of Rotation:

The **formula for 180 degree rotation** consists of a coordinate point in the x-y plane that has rotated by angle θ along the x-axis. The rotation formula used by the geometry rotation calculator is given below,

$$ X \;=\; x\; cos(\theta) + y\; sin(\theta) $$

$$ Y \;=\; -x\; sin(\theta) + y\; cos(\theta) $$

Whereas x is the coordinate of the x-axis y is the y coordinate value and θ is the angle of the rotational object.

## How to Calculate the Rotation?

For **calculating rotation**, the rotation solver uses several steps including the rule for rotation, and the process varies depending on the form of the rotational object in 2D or 3D space. Here’s a step-by-step guide for rotation problem calculation.

**For 2D Rotation**:

For 2D rotation, the rotation graph calculator determines the rotation point so, you need to find the rotating point (x,y) around the origin by an angle θ.

**Step 1**:

Identify the rotation function problem and the angle value theta function.

**Step 2**:

Convert the given angle into radians if it is present in degree.

$$ X \;=\; x\; cos(\theta) + y\; sin(\theta) $$

$$ Y \;=\; -x\; sin(\theta) + y\; cos(\theta) $$

$$ Radian \;=\; \frac{degrees \times180}{π} $$

**Step 3**:

Calculate the rotation problem after applying the formula of the rotation object.

**Step 4**:

Put the value in the above formula and simply it.

**Step 5**:

After simplification, you will give the solution of the new coordinates (x′,y′) that are the result of the rotation.

## Example of Rotation:

The rotation calculator helps you to determine the rotation point easily but its also essential to understand the step-by-step calculation process. So an **example** is given below,

**Example**:

Find a new representation of the equation 2x^{2} -xy + 2y^{2} - 30 = 0 after rotating through an angle of θ = 45°.

**Solution**:

The given equation and angle is,

$$ 2x^2 - xy + 2y^2 - 30 = 0,\; \theta \;=\; 45° $$

,

To find the value of x and y coordinates, the rotation formula is,

$$ X \;=\; x\; cos(\theta) + y\; sin(\theta) $$

$$ Y \;=\; -x\; sin(\theta) + y\; cos(\theta) $$

Put the x and y values in it,

$$ x \;=\; x’ cos \theta - y’ sin \theta $$

$$ x \;=\; x’ cos(45°) - y’ sin(45°) $$

Simplify the above expression for the solution of the x-coordinate value,

$$ x \;=\; x’ cos(45°) - y’ sin(45°) $$

$$ x \;=\; x’ \left( \frac{1}{\sqrt{2}} \right) - y’ \left(\frac{1}{\sqrt{2}} \right) $$

$$ x \;=\; \frac{x’ - y’}{\sqrt{2}} $$

For y coordinates values,

$$ y \;=\; x’\; sin(45°) + y’\; cos(45°) $$

Simplify it to get the solution,

$$ y \;=\; x’\; sin (45°) + y’\; cos(45°) $$

$$ y \;=\; x’\; \left(\frac{1}{\sqrt{2}} \right) + y’ \left(\frac{1}{\sqrt{2}} \right) $$

$$ y \;=\; \frac{x’ + y’}{\sqrt{2}} $$

## How to Use the Rotation Calculator?

The rotations calculator is an easy-to-use tool, so you can use it to evaluate the rotation of an object after simplifying it.

Before adding the input for the solution of given rotation problems in the rotation solver, you must follow some simple steps. These steps are:

- Enter the rotation function that you want to evaluate through the rotation geometry method.
- Recheck your input value for the solution of the rotation question before hitting the calculate button to start the calculation process.
- Click on the “
**Calculate**” button to get the desired result of your given rotation problem. - If you want to try out our rotation graph calculator to check its accuracy in solution, then you can use the load example.
- Click on the “Recalculate” button to get a new page for solving more rotation questions.

## Result from Rotations Calculator:

The rotating calculator gives you the **solution** to a given rotational function when you add the input value to it. It may contain as,

**Result Option**:

You can click on the result option and the rotate points calculator provides you with a solution of rotation geometry questions.

**Possible Step**:

When you click on the possible steps option, the rotating graph calculator provides you with the solution to the rotation problem.

## Benefits of Using Rotation Solver:

The rotate points calculator gives you many **benefits** whenever you use it to calculate rotation geometry problems and to get its solution. These benefits are:

- Our rotations calculator saves the time and effort that you consume in solving rotation questions and gets solutions in a few seconds.
- It is a free-of-cost tool that provides you with a solution for a given rotation problem to find the x and y coordinates value without paying a single penny.
- The rotating calculator is an adaptive tool that allows you to find the rotation of an object for finding x and y coordinates value.
- You can use this rotation graph calculator for practice to get familiar with this concept easily.
- It is a trustworthy tool that provides you with accurate solutions as per your input to calculate the rotation problem.
- The rotation calculator geometry is a learning tool so you can use it to teach your children, students, or professionals efficiently on an online platform.