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

• MHB
• GloriousGoats
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.
GloriousGoats
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.

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

Last edited:
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:

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

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.

## 1. How do I rotate a point by 75° counterclockwise?

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.

## 2. What is the difference between clockwise and counterclockwise rotation?

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.

## 3. Do I need any special tools or software to rotate a point by 75° counterclockwise?

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.

## 4. Can I rotate a point by any angle?

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.

## 5. Will rotating a point change its distance from the origin?

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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