A 36-degree counterclockwise rotation centered at the origin sends the point A=(6,3) to the image point A'. Use a rotation matrix to find coordinates for A'.

To find the coordinates of the image point \( A' \) after a 36-degree counterclockwise rotation centered at the origin, we can use the rotation matrix. The general formula for a counterclockwise rotation of an angle \( \theta \) about the origin is:

\[
\begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{bmatrix}
\]

For a 36-degree rotation, \( \theta = 36^\circ \). First, we need to find the values of \( \cos 36^\circ \) and \( \sin 36^\circ \).

From trigonometric tables or calculations, we have:

\[
\cos 36^\circ \approx 0.8090
\]
\[
\sin 36^\circ \approx 0.5878
\]

Using these values, the rotation matrix becomes:

\[
\begin{bmatrix}
0.8090 & -0.5878 \\
0.5878 & 0.8090
\end{bmatrix}
\]

Now, we multiply this matrix by the coordinates of point \( A = (6, 3) \).

\[
\begin{bmatrix}
0.8090 & -0.5878 \\
0.5878 & 0.8090
\end{bmatrix}
\begin{bmatrix}
6 \\
3
\end{bmatrix}
\]

Perform the matrix multiplication:

\[
\begin{bmatrix}
0.8090 \cdot 6 + (-0.5878) \cdot 3 \\
0.5878 \cdot 6 + 0.8090 \cdot 3
\end{bmatrix}
\]

Calculate each component:

\[
x' = 0.8090 \cdot 6 - 0.5878 \cdot 3 = 4.854 - 1.7634 = 3.0906
\]
\[
y' = 0.5878 \cdot 6 + 0.8090 \cdot 3 = 3.5268 + 2.427 = 5.9538
\]

So, the coordinates of the image point \( A' \) are approximately:

\[
A' \approx (3.0906, 5.9538)
\]

### Conclusion
The coordinates of the image point \( A' \) after a 36-degree counterclockwise rotation of point \( A = (6, 3) \) around the origin are approximately \( (3.0906, 5.9538) \).

Would you like more details or have any questions? Here are some related questions to explore further:

1. How do you derive the rotation matrix for

Learn how to find the coordinates of point A' after a 36-degree counterclockwise rotation centered at the origin using a rotation matrix. Includes step-by-step matrix multiplication and trigonometric calculations for cosine and sine values. Suitable for advanced high school students studying trigonometry and matrix transformations.

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