Find the polar coordinates of the points whose cartesian coordinates are
`(-3, 4)`

Answer

To find the polar coordinates of the point with Cartesian coordinates (-3, 4), we will follow these steps:

### Step 1: Identify the Cartesian Coordinates
The given Cartesian coordinates are:
- \( x = -3 \)
- \( y = 4 \)

### Step 2: Calculate the Radius \( r \)
The formula for calculating the radius \( r \) in polar coordinates is:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values of \( x \) and \( y \):
\[ r = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate the Angle \( \theta \)
The angle \( \theta \) can be found using the tangent function:
\[ \tan(\theta) = \frac{y}{x} \]
Substituting the values of \( y \) and \( x \):
\[ \tan(\theta) = \frac{4}{-3} \]
This gives us:
\[ \theta = \tan^{-1}\left(\frac{4}{-3}\right) \]
Since the point (-3, 4) is located in the second quadrant (where \( x < 0 \) and \( y > 0 \)), we need to adjust the angle:
\[ \theta = \pi + \tan^{-1}\left(\frac{4}{-3}\right) \]
To find the angle in the second quadrant, we can also express it as:
\[ \theta = \pi - \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 4: Final Polar Coordinates
Now we can express the polar coordinates as:
\[ (r, \theta) = \left(5, \pi - \tan^{-1}\left(\frac{4}{3}\right)\right) \]

### Conclusion
The polar coordinates of the point whose Cartesian coordinates are (-3, 4) are:
\[ \boxed{\left(5, \pi - \tan^{-1}\left(\frac{4}{3}\right)\right)} \]

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