Question

The cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \frac{1}{2}, 120^\circ \right) \text{ are}

• A. \( \left( \frac{1}{4}, -\frac{\sqrt{3}}{4} \right) \)
• B. \( \left( \frac{1}{4}, \frac{\sqrt{3}}{4} \right) \)
• C. \( \left( -\frac{1}{4}, -\frac{\sqrt{3}}{4} \right) \)
• D. \( \left( -\frac{1}{4}, \frac{\sqrt{3}}{4} \right) \)

Hint:

To convert from polar to Cartesian coordinates, use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \).

Answer 1

The Correct Option is D

Step 1: Use of Polar to Cartesian conversion formula.
The formula for converting polar coordinates \( (r, \theta) \) to Cartesian coordinates \( (x, y) \) is: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] Given \( r = \frac{1}{2} \) and \( \theta = 120^\circ \), we compute: \[ x = \frac{1}{2} \cos 120^\circ = \frac{1}{2} \times \left( -\frac{1}{2} \right) = -\frac{1}{4} \] \[ y = \frac{1}{2} \sin 120^\circ = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]
Step 2: Conclusion.
Thus, the Cartesian coordinates are \( \left( -\frac{1}{4}, \frac{\sqrt{3}}{4} \right) \), which corresponds to option (D).