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Properties of Modulus

Express the f...

Question

Express the following complex number in polar form and exponential form: $- 1 - \sqrt{3} i$

2 (c o s \frac{4 π}{3} + i s i n \frac{4 π}{3}); - 2 e^{i ⎛ ⎝ \frac{- 4 π}{3} ⎞ ⎠}

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- 2 (c o s \frac{4 π}{3} - i s i n \frac{4 π}{3}); 2 e^{i ⎛ ⎝ \frac{- 4 π}{3} ⎞ ⎠}

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2 (c o s \frac{2 π}{3} - i s i n \frac{2 π}{3}); 2 e^{i ⎛ ⎝ \frac{- 2 π}{3} ⎞ ⎠}

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- 2 (c o s \frac{2 π}{3} - i s i n \frac{2 π}{3}); - 2 e^{i ⎛ ⎝ \frac{- 2 π}{3} ⎞ ⎠}

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Solution

The correct option is D $2 (c o s \frac{2 π}{3} - i s i n \frac{2 π}{3}); 2 e^{i ⎛ ⎝ \frac{- 2 π}{3} ⎞ ⎠}$
$z = (- 1 - \sqrt{3} i)$
$= 2 (\frac{- 1 - \sqrt{3} i}{2})$
$= 2 (c o s (\frac{π}{3} - π) + i s i n (\frac{π}{3} - π))$
$= 2 (c o s (\frac{- 2 π}{3}) + i s i n (\frac{- 2 π}{3}))$
$= 2 e^{i \frac{- 2 π}{3}}$

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Similar questions

a = c o s x + c o s (x + \frac{2 π}{3}) + c o s (x + \frac{4 π}{3})

b = s i n x + s i n (x + \frac{2 π}{3}) + s i n (x + \frac{4 π}{3})

Δ

Θ, - 5 + 4 s i n Θ

B = (3 + 4 c o s (Θ + \frac{2 π}{3}), - 5 + 4 s i n (Θ + \frac{2 π}{3}))

C = (3 + 4 c o s (Θ + \frac{4 π}{3}), - 5 + 4 s i n (Θ + \frac{4 π}{3}))

Δ

Θ, - 5 + 4 s i n Θ

B = (3 + 4 c o s (Θ + \frac{2 π}{3}), - 5 + 4 s i n (Θ + \frac{2 π}{3}))

C = (3 + 4 c o s (Θ + \frac{4 π}{3}), - 5 + 4 s i n (Θ + \frac{4 π}{3}))

{sin}^{3} x + {sin}^{3} (x + \frac{2 π}{3}) + {sin}^{3} (x + \frac{4 π}{3}) = 3 sin x sin (x + \frac{2 π}{3}) \cdot sin (x + \frac{4 π}{3})

a + b + c = 0

a^{3} + b^{3} + c^{3} = 3 a b c

- 1 - \sqrt{3} i

{cos}^{3} α + {cos}^{3} (α + \frac{2 π}{3}) + {cos}^{3} (α + \frac{4 π}{3}) = 3 cos α cos (α + \frac{2 π}{3}) cos (α + \frac{4 π}{3})

a + b + c = 0 \Leftrightarrow a^{3} + b^{3} + c^{3} = 3 a b c

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