The complex number z = i - 1 cos 3 + i sin 3 is equal to [2023]

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Q.
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]

1 $\sqrt{2} i (\cos \frac{5 π}{12} - i \sin \frac{5 π}{12})$

2 $\cos \frac{π}{12} - i \sin \frac{π}{12}$

3 $\sqrt{2} (\cos \frac{π}{12} + i \sin \frac{π}{12})$

4 $\sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

Ans.
(4)

$z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}} = \frac{\sqrt{2} e^{i \frac{3 π}{4}}}{e^{i \frac{π}{3}}}$

$= \sqrt{2} e^{i (\frac{3 π}{4} - \frac{π}{3})} = \sqrt{2} e^{i (\frac{5 π}{12})} = \sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

Solution

Q.
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]

1 $\sqrt{2} i (\cos \frac{5 π}{12} - i \sin \frac{5 π}{12})$

2 $\cos \frac{π}{12} - i \sin \frac{π}{12}$

3 $\sqrt{2} (\cos \frac{π}{12} + i \sin \frac{π}{12})$

4 $\sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

Ans.
(4)

$z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}} = \frac{\sqrt{2} e^{i \frac{3 π}{4}}}{e^{i \frac{π}{3}}}$

$= \sqrt{2} e^{i (\frac{3 π}{4} - \frac{π}{3})} = \sqrt{2} e^{i (\frac{5 π}{12})} = \sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

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Solution

Q. The complex number z=i-1cosπ3+isinπ3 is equal to [2023]

1 2i(cos5π12-isin5π12)

2 cosπ12-isinπ12

3 2(cosπ12+isinπ12)

4 2(cos5π12+isin5π12)

Ans. (4) z=i-1cosπ3+isinπ3 =2ei3π4eiπ3 =2ei(3π4-π3) =2ei(5π12) =2(cos5π12+isin5π12)

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Q.
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]

1 $\sqrt{2} i (\cos \frac{5 π}{12} - i \sin \frac{5 π}{12})$

2 $\cos \frac{π}{12} - i \sin \frac{π}{12}$

3 $\sqrt{2} (\cos \frac{π}{12} + i \sin \frac{π}{12})$

4 $\sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

Ans.
(4)

$z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}} = \frac{\sqrt{2} e^{i \frac{3 π}{4}}}{e^{i \frac{π}{3}}}$

$= \sqrt{2} e^{i (\frac{3 π}{4} - \frac{π}{3})} = \sqrt{2} e^{i (\frac{5 π}{12})} = \sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$