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Question: If \(|z| = 4\) and arg \(z = \frac{5\pi}{6},\) then \(z =\)...

If ∣z∣=4|z| = 4 and arg z=5π6,z = \frac{5\pi}{6}, then z=z =

A

23−2i2\sqrt{3} - 2i

B

23+2i2\sqrt{3} + 2i

C

−23+2i- 2\sqrt{3} + 2i

D

−3+i- \sqrt{3} + i

Answer

−23+2i- 2\sqrt{3} + 2i

Explanation

Solution

Sol. ∣z∣=4|z| = 4 and arg z=5π6=150ºz = \frac{5\pi}{6} = 150º,

Let z=x+iyz = x + iy, then ∣z∣=r=x2+y2=4|z| = r = \sqrt{x^{2} + y^{2}} = 4 and

θ=5π6=150º\theta = \frac{5\pi}{6} = 150º

∴x=rcos⁡θ=4cos⁡150º=−23\therefore x = r\cos{}\theta = 4\cos 150º = - 2\sqrt{3}and y=rsin⁡θ=4sin⁡150º=4⋅12=2.y = r\sin\theta = 4\sin 150º = 4 \cdot \frac{1}{2} = 2.

∴z=x+iy=−23+2i.\therefore z = x + iy = - 2\sqrt{3} + 2i.

Trick: Since arg z=5π6=150º,z = \frac{5\pi}{6} = 150º, here the complex number must lie in second quadrant, so (1) and (2) rejected. Also ∣z∣=4|z| = 4, which satisfies (3) only.