MHB Rotate Point p: How to Rotate by 75° Counterclockwise

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To rotate point p=(3,3√3) counterclockwise by 75 degrees about the origin, one can utilize the rotation matrix, which incorporates cosine and sine functions of the angle. The approach involves calculating the distance from the origin to point p and determining the angle of inclination using arctangent. By adding 75 degrees to the angle of inclination, the new coordinates can be derived using the formulas for cosine and sine. Visualizing the rotation by sketching the point and its new position can simplify the process. Ultimately, understanding the geometric implications can aid in solving the problem without complex formulas.
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My problem reads as follows: Point p=(3,3√3) is rotated counterclockwise about the origin by 75 degrees. What are the coordinates after this rotation?
I have no idea how to rotate a point, let alone by 75 degrees.
 
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Hint 1: The matrix
$$\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$$
represents a counterclockwise rotation of $\theta$ about the origin.

Hint 2: $75^\circ\ =\ 45^\circ+30^\circ$.
 
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The way I would approach this is:

1.) Find the distance \(r\) from the origin to the given point, and find the angle of inclination \(\theta=\arctan(m)\), where \(m\) is the slope of the line through the origin and the given point. Let \(\alpha=\theta+75^{\circ}\).

2.) The rotated point will then be:

$$(r\cos(\alpha),r\sin(\alpha))$$
 
GloriousGoats said:
My problem reads as follows: Point p=(3,3√3) is rotated counterclockwise about the origin by 75 degrees. What are the coordinates after this rotation?
I have no idea how to rotate a point, let alone by 75 degrees.

If you think about where the point p is and where it will be after the rotation, you won't need any rotation formulas or addition formulas, just the basic angles. Draw a picture.
 
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