Math Problem Statement
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'.
Solution
To find the coordinates of the image point 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 about the origin is:
\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 forAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Rotation matrices
Formulas
Rotation matrix formula for counterclockwise rotation
Theorems
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Suitable Grade Level
Advanced High School