Rotation Math Example 2

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Example 2

medium
Rotate the point B(4,0)B(4, 0) by 60°60° counterclockwise about the origin. Give exact coordinates.

Solution

  1. 1
    Step 1: Use the rotation formula: (x′,y′)=(xcos⁡θ−ysin⁡θ, xsin⁡θ+ycos⁡θ)(x', y') = (x\cos\theta - y\sin\theta,\ x\sin\theta + y\cos\theta).
  2. 2
    Step 2: With x=4x=4, y=0y=0, θ=60°\theta=60°: cos⁡60°=12\cos60°=\tfrac{1}{2}, sin⁡60°=32\sin60°=\tfrac{\sqrt{3}}{2}.
  3. 3
    Step 3: x′=4⋅12−0=2x' = 4\cdot\tfrac{1}{2} - 0 = 2.
  4. 4
    Step 4: y′=4⋅32+0=23y' = 4\cdot\tfrac{\sqrt{3}}{2} + 0 = 2\sqrt{3}.
  5. 5
    Step 5: B′=(2, 23)B' = (2,\ 2\sqrt{3}).

Answer

B′=(2, 23)B' = (2,\ 2\sqrt{3})
The general rotation matrix is (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}. For θ=60°\theta = 60°, the exact values are cos⁡60°=1/2\cos60°=1/2 and sin⁡60°=3/2\sin60°=\sqrt{3}/2. Knowing these exact values avoids decimal rounding errors.

About Rotation

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

Learn more about Rotation →

More Rotation Examples

Example 1 easy

Rotate the point [formula] by [formula] counterclockwise about the origin. Where does A end up?

Example 3 easy

What is the image of point [formula] after a [formula] rotation about the origin?

Example 4 hard

Triangle with vertices [formula], [formula], [formula] is rotated [formula] CCW about the point [for

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