To determine the new coordinates after rotating a point around the origin, apply the rotation formulas using the angle of rotation, θ This process adjusts each coordinate based on trigonometric functions for precise new positions.

In geometry, rotation of coordinates is a transformational method that repositions points or shapes around a fixed center. This rotation calculator is really useful for speedily finding the new coordinates after rotating a point by a specific angle around the origin.

Generally, this tool has various applications in mathematics, physics, engineering, and computer graphics. To say the least, this tool is ideal for tasks that involve object rotation, angular adjustments, and transformations in different coordinate systems.

**Formula:**

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

Variable | Description |
---|---|

$X$ | New x-coordinate after rotation |

$Y$ | New y-coordinate after rotation |

$x$ | Original x-coordinate |

$y$ | Original y-coordinate |

$\theta$ | Angle of rotation (in degrees) |

**Solved Calculations:**

**Example 1:**

Step | Calculation |
---|---|

Original Point (x, y) | (3, 4) |

Angle of Rotation ($\theta$) | 90° |

New X Calculation | $3 \cos(90°) + 4 \sin(90°)$ |

Result for X | $0 + 4 = 4$ |

New Y Calculation | $-3 \sin(90°) + 4 \cos(90°)$ |

Result for Y | $-3 + 0 = -3$ |

**Answer**: The new coordinates are (4, -3).

**Example 2:**

Step | Calculation |
---|---|

Original Point (x, y) | (5, 2) |

Angle of Rotation ($\theta$) | 180° |

New X Calculation | $5 \cos(180°) + 2 \sin(180°)$ |

Result for X | $-5 + 0 = -5$ |

New Y Calculation | $-5 \sin(180°) + 2 \cos(180°)$ |

Result for Y | $0 – 2 = -2$ |

**Answer**: The new coordinates are (-5, -2).

**What is a Rotation Calculator?**