Find the cartesian coordinates of the points whose polar coordinates are
`(5 sqrt(2), (pi)/(4))`

Answer

To convert the given polar coordinates \((5\sqrt{2}, \frac{\pi}{4})\) into Cartesian coordinates \((x, y)\), we can follow these steps:

### Step 1: Identify the values of \(R\) and \(\theta\)
The polar coordinates are given as:
- \(R = 5\sqrt{2}\)
- \(\theta = \frac{\pi}{4}\)

### Step 2: Use the conversion formulas
The conversion from polar coordinates to Cartesian coordinates is done using the following formulas:
- \(x = R \cos(\theta)\)
- \(y = R \sin(\theta)\)

### Step 3: Calculate \(x\)
Substituting the values of \(R\) and \(\theta\) into the formula for \(x\):
\[ x = 5\sqrt{2} \cos\left(\frac{\pi}{4}\right) \]
We know that \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\):
\[ x = 5\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 5 \]

### Step 4: Calculate \(y\)
Now, substituting the values into the formula for \(y\):
\[ y = 5\sqrt{2} \sin\left(\frac{\pi}{4}\right) \]
We know that \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\):
\[ y = 5\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 5 \]

### Step 5: Write the final Cartesian coordinates
Thus, the Cartesian coordinates corresponding to the given polar coordinates are:
\[ (x, y) = (5, 5) \]

### Final Answer:
The Cartesian coordinates are \((5, 5)\).

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