To rotate a point $P = (2, 3)$ by 35 degrees around the origin in a 2D plane, we can use the following rotation formulas:

$x' = x \cos \theta - y \sin \theta$ $y' = x \sin \theta + y \cos \theta$

Where:

• $(x, y)$ are the original coordinates of the point.
• $(x', y')$ are the new coordinates after rotation.
• $\theta$ is the angle of rotation (in radians).
• $\cos$ and $\sin$ are the cosine and sine trigonometric functions, respectively.

Step 1: Convert the angle to radians

Since trigonometric functions in most calculators and computer systems use radians, we need to convert 35 degrees to radians.

$\theta = 35^\circ \times \frac{\pi}{180^\circ} = \frac{35\pi}{180} \approx 0.6109 \text{ radians}$

Step 2: Apply the rotation formulas

Using $x = 2$, $y = 3$, and $\theta \approx 0.6109$ radians:

$x' = 2 \cos(0.6109) - 3 \sin(0.6109)$ $y' = 2 \sin(0.6109) + 3 \cos(0.6109)$

Step 3: Compute the cosine and sine of 0.6109 radians

$\cos(0.6109) \approx 0.8192$ $\sin(0.6109) \approx 0.5736$

Step 4: Calculate the new coordinates

Substituting these values:

$x' = 2(0.8192) - 3(0.5736) = 1.6384 - 1.7208 = -0.0824$ $y' = 2(0.5736) + 3(0.8192) = 1.1472 + 2.4576 = 3.6048$

Final Answer

After rotating the point $P = (2, 3)$ by 35 degrees, the new coordinates are approximately:

$P' = (-0.0824, 3.6048)$

Would you like more details on any of these steps, or do you have another question?

5 Related Questions:

1. How do you rotate a point around an arbitrary point instead of the origin?
2. What are the implications of rotating a point by 90, 180, and 270 degrees?
3. How would you rotate a point in 3D space?
4. How do you find the inverse rotation of a point?
5. How does matrix multiplication relate to 2D point rotation?

Tip:

Always double-check your trigonometric calculations when rotating points, especially when converting between degrees and radians.