Parametric equation of the circle `x^(2)+y^(2)=16` are

Options

A. `x=4cos theta, y=4sin theta`
B. `x=4 cos theta,y=4tan theta`
C. `x=4 cos h theta , y=4 sin h theta`
D. `x=4 sec theta, y=4 tan theta`

Answer

To find the parametric equations of the circle given by the equation \(x^2 + y^2 = 16\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the standard form of the circle**:
The equation of the circle can be compared to the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

2. **Determine the center and radius**:
From the equation \(x^2 + y^2 = 16\):
- We can see that \(h = 0\) and \(k = 0\) (the center is at the origin \((0, 0)\)).
- The radius \(r\) can be found by taking the square root of 16, which gives \(r = 4\).

3. **Set up the parametric equations**:
To express the coordinates \(x\) and \(y\) in terms of a parameter \(\theta\), we use the following parametric equations for a circle:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]

4. **Substitute the radius into the equations**:
Since we found that \(r = 4\), we substitute this value into the parametric equations:
\[ x = 4 \cos(\theta) \]
\[ y = 4 \sin(\theta) \]

5. **Final parametric equations**:
Therefore, the parametric equations of the circle \(x^2 + y^2 = 16\) are:
\[ x = 4 \cos(\theta), \quad y = 4 \sin(\theta) \]

### Conclusion:
The correct option for the parametric equations of the circle is:
- \(x = 4 \cos(\theta)\)
- \(y = 4 \sin(\theta)\)