Math Problem Statement
Determine the coordinates of the point resulting from a rotation of centred at the origin and starting from the point (3, 0).
Solution
To determine the coordinates of a point resulting from a rotation centered at the origin, we can use the standard rotation formulas in the coordinate plane.
Rotation Formulas
Given a point and rotating it counterclockwise by an angle around the origin, the new coordinates are given by:
Here, is the starting point, and is the rotated point.
Starting Point
We are starting with the point , so:
Coordinates after Rotation
-
Substituting and into the formulas:
Simplifies to:
General Result
The coordinates of the rotated point are:
Here, is the angle of rotation.
Examples:
To provide more clarity, let’s examine the coordinates for common angles:
-
Rotation by ():
- ,
- Coordinates:
-
Rotation by ():
- ,
- Coordinates:
-
Rotation by ():
- ,
- Coordinates:
-
Rotation by ():
- ,
- Coordinates:
If you know the angle of rotation, you can substitute it into the general formula .
Let me know if you'd like further details or additional examples!
Related Questions:
- What are the coordinates of a point after a rotation?
- How do you derive the rotation formulas mathematically?
- What happens if the rotation is clockwise instead of counterclockwise?
- How do you rotate a point around a center other than the origin?
- How do you generalize rotations in 3D space?
Tip:
To visualize rotations, sketch the initial point, draw the angle of rotation, and use trigonometric functions to determine the new coordinates.
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Math Problem Analysis
Mathematical Concepts
Geometry
Trigonometry
Rotation
Formulas
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Theorems
Rotation Theorem (Counterclockwise Rotation around the Origin)
Suitable Grade Level
Grades 9-11
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