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 P(x,y)P(x, y) and rotating it counterclockwise by an angle θ\theta around the origin, the new coordinates P(x,y)P'(x', y') are given by:

x=xcos(θ)ysin(θ),y=xsin(θ)+ycos(θ)x' = x\cos(\theta) - y\sin(\theta), \quad y' = x\sin(\theta) + y\cos(\theta)

Here, (x,y)(x, y) is the starting point, and (x,y)(x', y') is the rotated point.


Starting Point

We are starting with the point (3,0)(3, 0), so:

x=3andy=0x = 3 \quad \text{and} \quad y = 0


Coordinates after Rotation

  1. Substituting x=3x = 3 and y=0y = 0 into the formulas: x=3cos(θ)0sin(θ)andy=3sin(θ)+0cos(θ)x' = 3\cos(\theta) - 0\sin(\theta) \quad \text{and} \quad y' = 3\sin(\theta) + 0\cos(\theta)

    Simplifies to:

    x=3cos(θ),y=3sin(θ)x' = 3\cos(\theta), \quad y' = 3\sin(\theta)


General Result

The coordinates of the rotated point are:

(3cos(θ),3sin(θ))\boxed{(3\cos(\theta), \, 3\sin(\theta))}

Here, θ\theta is the angle of rotation.


Examples:

To provide more clarity, let’s examine the coordinates for common angles:

  1. Rotation by 9090^\circ (θ=π2\theta = \frac{\pi}{2}):

    • cos(π2)=0\cos\left(\frac{\pi}{2}\right) = 0, sin(π2)=1\sin\left(\frac{\pi}{2}\right) = 1
    • Coordinates: (30,31)=(0,3)(3 \cdot 0, 3 \cdot 1) = (0, 3)
  2. Rotation by 180180^\circ (θ=π\theta = \pi):

    • cos(π)=1\cos(\pi) = -1, sin(π)=0\sin(\pi) = 0
    • Coordinates: (31,30)=(3,0)(3 \cdot -1, 3 \cdot 0) = (-3, 0)
  3. Rotation by 270270^\circ (θ=3π2\theta = \frac{3\pi}{2}):

    • cos(3π2)=0\cos\left(\frac{3\pi}{2}\right) = 0, sin(3π2)=1\sin\left(\frac{3\pi}{2}\right) = -1
    • Coordinates: (30,31)=(0,3)(3 \cdot 0, 3 \cdot -1) = (0, -3)
  4. Rotation by 360360^\circ (θ=2π\theta = 2\pi):

    • cos(2π)=1\cos(2\pi) = 1, sin(2π)=0\sin(2\pi) = 0
    • Coordinates: (31,30)=(3,0)(3 \cdot 1, 3 \cdot 0) = (3, 0)

If you know the angle of rotation, you can substitute it into the general formula (3cos(θ),3sin(θ))(3\cos(\theta), 3\sin(\theta)).

Let me know if you'd like further details or additional examples!


Related Questions:

  1. What are the coordinates of a point after a 4545^\circ rotation?
  2. How do you derive the rotation formulas mathematically?
  3. What happens if the rotation is clockwise instead of counterclockwise?
  4. How do you rotate a point around a center other than the origin?
  5. 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