Find polar coordinate of `(1,sqrt(3))`

Answer

To find the polar coordinates of the point \((1, \sqrt{3})\), we need to determine two values: \(r\) (the distance from the origin) and \(\theta\) (the angle with respect to the positive x-axis).

### Step-by-Step Solution:

1. **Identify the Cartesian Coordinates**:
The given point is \((x, y) = (1, \sqrt{3})\).

2. **Calculate the Distance \(r\)**:
The distance \(r\) from the origin to the point can be calculated using the formula:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values of \(x\) and \(y\):
\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

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 \(x\):
\[ \tan(\theta) = \frac{\sqrt{3}}{1} = \sqrt{3} \]
The angle \(\theta\) for which \(\tan(\theta) = \sqrt{3}\) is:
\[ \theta = \frac{\pi}{3} \text{ radians} \quad \text{(or 60 degrees)} \]

4. **Write the Polar Coordinates**:
The polar coordinates of the point \((1, \sqrt{3})\) are:
\[ (r, \theta) = (2, \frac{\pi}{3}) \]

### Final Answer:
The polar coordinates of the point \((1, \sqrt{3})\) are \((2, \frac{\pi}{3})\).