The cartesian co-ordinates of a point, whose polar co-ordinates are `((3)/(4), 135^(@))` are

Options

A. `((-3)/(4sqrt(2)), (-3)/(4sqrt(2)))`
B. `((3)/(4sqrt(2)), (3)/(4sqrt(2)))`
C. `((-3)/(4sqrt(2)), (3)/(4sqrt(2)))`
D. `((3)/(4sqrt(2)), (-3)/(4sqrt(2)))`

Answer

To find the Cartesian coordinates of a point given its polar coordinates \((\frac{3}{4}, 135^\circ)\), we will use the following formulas:

1. \(x = R \cos(\theta)\)
2. \(y = R \sin(\theta)\)

Where:
- \(R\) is the radial coordinate (distance from the origin).
- \(\theta\) is the angular coordinate (angle from the positive x-axis).

### Step 1: Identify the values of \(R\) and \(\theta\)
Given polar coordinates:
- \(R = \frac{3}{4}\)
- \(\theta = 135^\circ\)

### Step 2: Convert degrees to radians (if necessary)
In this case, we can directly use the degrees since trigonometric functions can handle degrees. However, if needed:
\[ 135^\circ = \frac{3\pi}{4} \text{ radians} \]

### Step 3: Calculate \(x\)
Using the formula for \(x\):
\[ x = R \cos(\theta) = \frac{3}{4} \cos(135^\circ) \]
Now, we know that:
\[ \cos(135^\circ) = -\frac{1}{\sqrt{2}} \text{ (since 135° is in the second quadrant)} \]
Thus,
\[ x = \frac{3}{4} \left(-\frac{1}{\sqrt{2}}\right) = -\frac{3}{4\sqrt{2}} = -\frac{3\sqrt{2}}{8} \text{ (after rationalizing the denominator)} \]

### Step 4: Calculate \(y\)
Using the formula for \(y\):
\[ y = R \sin(\theta) = \frac{3}{4} \sin(135^\circ) \]
And we know that:
\[ \sin(135^\circ) = \frac{1}{\sqrt{2}} \text{ (since 135° is in the second quadrant)} \]
Thus,
\[ y = \frac{3}{4} \left(\frac{1}{\sqrt{2}}\right) = \frac{3}{4\sqrt{2}} = \frac{3\sqrt{2}}{8} \text{ (after rationalizing the denominator)} \]

### Step 5: Write the Cartesian coordinates
Now we have:
\[ x = -\frac{3\sqrt{2}}{8}, \quad y = \frac{3\sqrt{2}}{8} \]
So, the Cartesian coordinates are:
\[ \left(-\frac{3\sqrt{2}}{8}, \frac{3\sqrt{2}}{8}\right) \]

### Final Answer
The Cartesian coordinates of the point are:
\[ \left(-\frac{3\sqrt{2}}{8}, \frac{3\sqrt{2}}{8}\right) \]