The polar co-ordinates of the point whose cartesian co-ordinates are `(1, sqrt(3))`, are

Options

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

Answer

To find the polar coordinates of the point whose Cartesian coordinates are \((1, \sqrt{3})\), we will follow these steps:

### Step 1: Calculate the value of \( r \)
The formula for \( r \) in polar coordinates is given by:
\[ r = \sqrt{x^2 + y^2} \]
Here, \( x = 1 \) and \( y = \sqrt{3} \). Plugging in these values, we get:
\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 2: Calculate the value of \( \theta \)
The angle \( \theta \) can be found using the tangent function:
\[ \tan(\theta) = \frac{y}{x} \]
Substituting the values of \( y \) and \( x \):
\[ \tan(\theta) = \frac{\sqrt{3}}{1} = \sqrt{3} \]
To find \( \theta \), we take the inverse tangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
The angle whose tangent is \( \sqrt{3} \) is \( \frac{\pi}{3} \) radians (or 60 degrees).

### Step 3: Write the polar coordinates
Now that we have both \( r \) and \( \theta \), we can express the polar coordinates as:
\[ (r, \theta) = (2, \frac{\pi}{3}) \]

Thus, the polar coordinates of the point whose Cartesian coordinates are \((1, \sqrt{3})\) are \((2, \frac{\pi}{3})\).

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