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: Read the polar information.
The complex number has distance from origin $r=2$ and argument $\theta=\frac{5\pi}{6}$.
Step 2: Write the polar form.
$z=r(\cos\theta+i\sin\theta)=2\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$.
Step 3: Locate the angle.
$\frac{5\pi}{6}$ is in the second quadrant, where cosine is negative and sine is positive. The reference angle is $\frac{\pi}{6}$.
Step 4: Get the exact values.
$\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$ and $\sin\frac{5\pi}{6}=\frac{1}{2}$.
Step 5: Substitute and multiply.
$z=2\left(-\frac{\sqrt{3}}{2}+i\cdot\frac{1}{2}\right)=-\sqrt{3}+i$.
Step 6: State the answer.
So $z=-\sqrt{3}+i$, which is option (4). \[ \boxed{z=-\sqrt{3}+i} \]