Question

The complex number with argument $\frac{5\pi}{6}$ at a distance of 2 units from the origin is

• A. $\sqrt{3}-i$
• B. $\sqrt{3}+i$
• C. $-\sqrt{3}-i$
• D. $-\sqrt{3}+i$

Hint:

You can often solve this type of question in 5 seconds just by checking signs! An argument of $\frac{5\pi}{6}$ places the point firmly in the 2nd quadrant. In the 2nd quadrant, the real part ($a$) must be negative and the imaginary part ($b$) must be positive. Only $-\sqrt{3}+i$ fits this $(-, +)$ coordinate pattern!

Answer 1

The Correct Option is D

Step 1: Use polar to rectangular form.
$z = r(\cos\theta + i\sin\theta)$ with $r = 2$ and $\theta = \tfrac{5\pi}{6}$.

Step 2: Find the trig values.
The angle is in the second quadrant, so $\cos\tfrac{5\pi}{6} = -\tfrac{\sqrt3}{2}$ and $\sin\tfrac{5\pi}{6} = \tfrac12$.

Step 3: Multiply out.
$$z = 2\left(-\tfrac{\sqrt3}{2} + i\tfrac12\right) = -\sqrt3 + i$$
\[ \boxed{-\sqrt3 + i} \]