Find the cartesian coordinates of the points whose polar coordinates are
`(5, pi - tan^(-1)((4)/(3)))`

Answer

To find the Cartesian coordinates of the point whose polar coordinates are given as \( (5, \pi - \tan^{-1}(\frac{4}{3})) \), we will follow these steps:

### Step 1: Identify the values of \( r \) and \( \theta \)
From the polar coordinates, we have:
- \( r = 5 \)
- \( \theta = \pi - \tan^{-1}(\frac{4}{3}) \)

### Step 2: Calculate \( \tan(\theta) \)
Using the identity for tangent, we have:
\[ \tan(\theta) = \tan(\pi - \tan^{-1}(\frac{4}{3})) = -\tan(\tan^{-1}(\frac{4}{3})) = -\frac{4}{3} \]
This means that:
\[ \cot(\theta) = -\frac{3}{4} \]

### Step 3: Determine the sides of the right triangle
From the cotangent value, we can identify the sides of the right triangle:
- Base (adjacent side) = -4 (since cotangent is negative in the second quadrant)
- Perpendicular (opposite side) = 3

### Step 4: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 5: Calculate \( \cos(\theta) \) and \( \sin(\theta) \)
Now, we can find \( \cos(\theta) \) and \( \sin(\theta) \):
\[ \cos(\theta) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{-4}{5} \]
\[ \sin(\theta) = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{3}{5} \]

### Step 6: Convert to Cartesian coordinates
Using the formulas:
\[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \]
Substituting the values:
\[ x = 5 \cdot \left(-\frac{4}{5}\right) = -4 \]
\[ y = 5 \cdot \left(\frac{3}{5}\right) = 3 \]

### Step 7: Write the final Cartesian coordinates
Thus, the Cartesian coordinates are:
\[ (-4, 3) \]