- #1

GloriousGoats

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I have no idea how to rotate a point, let alone by 75 degrees.

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In summary, the given point p=(3,3√3) is rotated counterclockwise about the origin by 75 degrees. To find the coordinates after this rotation, we can use the distance and angle of inclination of the point from the origin. The rotated point will be located at (rcos(θ+75°), rsin(θ+75°)), where r is the distance from the origin and θ is the angle of inclination. No additional rotation formulas or addition formulas are needed, just a basic understanding of angles.

- #1

GloriousGoats

- 1

- 0

I have no idea how to rotate a point, let alone by 75 degrees.

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- #2

Olinguito

- 239

- 0

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: $75^\circ\ =\ 45^\circ+30^\circ$.

$$\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$$

represents a counterclockwise rotation of $\theta$ about the origin.

Hint 2: $75^\circ\ =\ 45^\circ+30^\circ$.

Last edited:

- #3

MarkFL

Gold Member

MHB

- 13,288

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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:

\(\displaystyle (r\cos(\alpha),r\sin(\alpha))\)

- #4

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- 775

GloriousGoats said:

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.

To rotate a point by 75° counterclockwise, you can use the following formula:

x' = x*cos(75°) - y*sin(75°)

y' = x*sin(75°) + y*cos(75°)

This will give you the new coordinates of the rotated point, where (x,y) are the original coordinates and (x',y') are the new coordinates.

A clockwise rotation is a rotation in the direction that the hands of a clock move, while a counterclockwise rotation is in the opposite direction.

In terms of coordinates, a clockwise rotation would result in a negative angle, while a counterclockwise rotation would have a positive angle.

No, you do not need any special tools or software to rotate a point by 75° counterclockwise. You can use a calculator or a programming language to perform the necessary calculations.

Yes, you can rotate a point by any angle. The formula for rotating a point can be adjusted to work for any angle, as long as you know the values of sine and cosine for that angle.

Yes, rotating a point will change its distance from the origin. The new coordinates of the rotated point will be at a different distance from the origin, unless the angle of rotation is a multiple of 90 degrees.

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