Given a 2D point P = (2, 3), Rotate the point by rotate the point by 35 degrees.

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.

Learn how to rotate a point (2, 3) by 35 degrees using trigonometric formulas. This problem involves applying rotation formulas in 2D coordinate geometry, converting degrees to radians, and calculating new coordinates post-rotation.

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