The point (3, 2) is rotated 30 degrees about the origin. What are the polar coordinates of this after rotation?
Am I right in saying it's (root13, 30)? The angle rotated and the radius calculated as root(3^2 + 2^2)? I can convert between polar and cartesian, but I don't recall how to do rotation.
you can also do the rotation simply by adjusting the angle. Rotate (√13,30) by 30, and you end up at (√13,60).
Now just convert that back to rectangular coordinates.
To find the polar coordinates after rotating the point (3, 2) by 30 degrees about the origin, you need to follow these steps:
1. Calculate the angle between the positive x-axis and the line connecting the origin and the point (3, 2). You can use the arctan function to find this angle:
θ = arctan(y/x) = arctan(2/3)
2. Add the rotation angle of 30 degrees to the calculated angle θ:
θ' = θ + 30°
3. Calculate the distance (radius) from the origin to the point (3, 2) using the Pythagorean theorem:
r = √(x^2 + y^2) = √(3^2 + 2^2)
4. Use the polar coordinates formula to find the polar coordinates (r', θ') after rotation:
r' = r
θ' = θ'
So the polar coordinates after rotating the point (3, 2) by 30 degrees about the origin are (√13, arctan(2/3) + 30°).
To find the polar coordinates of a point after rotating it, you can use the following steps:
1. Convert the given Cartesian coordinates (x, y) to polar coordinates (r, θ).
- The radius (r) can be calculated using the formula: r = √(x^2 + y^2).
- The angle (θ) can be calculated using the formula: θ = arctan(y/x).
In this case, for the point (3, 2):
- r = √(3^2 + 2^2) = √(9 + 4) = √13.
- θ = arctan(2/3) = approximately 33.69 degrees.
2. Apply the rotation to the angle θ as follows:
- Rotate the angle θ by the desired rotation angle (in this case, 30 degrees).
- To rotate counterclockwise, add the rotation angle to θ. To rotate clockwise, subtract the rotation angle from θ.
In this case, rotating 30 degrees counterclockwise:
- θ_after_rotation = θ + 30 degrees = 33.69 + 30 = 63.69 degrees.
3. Finally, express the new polar coordinates (r_after_rotation, θ_after_rotation).
For this point after rotation by 30 degrees:
- r_after_rotation = √13 (the radius remains the same).
- θ_after_rotation = 63.69 degrees.
So, the polar coordinates of the point (3, 2) after rotating it 30 degrees counterclockwise about the origin are (√13, 63.69 degrees).
The rotation matrix about the origin is
R=
| cos(t) -sin(t) |
| sin(t) +cos(t) |
Any rotation through angle t of a point X(x,y) is then given by
RX
Put t=pi/6=30°
R=
| 0.8660 -0.5000 |
| 0.5000 +0.8660 |
and X(3,2)
Do the matrix multiplication:
RX=(1.598,3.232)
Note that a rotation about the origin does not change the radius, or distance from the origin, thus
2²+3²=13
1.598²+3.232²=13