The continued product of the four values of -cos-3- i sin-3-3 - 4A- -1B- 1C- -2D- 2

Given: [cos(π3) nbsp;+ isin(π3)]^34 Let z=(cosπ + i sinπ)^34 ⇒ z= nbsp;(cosπ + i sinπ)^1/4 ⇒ z^4=cosπ + i sinπ ⇒ z^4 = 1 If z_1,z_2,z_3,z_4 are the roots, ...

You visited us 3 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Geometrical Representation of Argument and Modulus

The continued...

Question

The continued product of the four values of ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$ is

- 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

1

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2

No worries! We‘ve got your back. Try BYJU‘S free classes today!

- 2

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $1$
We have, ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$
$= (cos π + i sin π)^{1 / 4}$
Let $z = (cos π + i sin π)^{1 / 4}$
$∴ z^{4} = cos π + i sin π = - 1$
If $z_{1}, z_{2}, z_{3}, z_{4}$ are the roots then
$z_{1} z_{2} z_{3} z_{4} = - 1 (- 1)^{4 - 1} = 1$

Suggest Corrections

Similar questions

Q. The value of

\sqrt{3 - 2 \sqrt{2}}

is

(a)

\sqrt{2} - 1

(b)

\sqrt{2} + 1

(c)

\sqrt{3} - \sqrt{2}

(d)

\sqrt{3} + \sqrt{2}

Q. The value of

{(\frac{1 + c o s \frac{π}{2} - i s i n \frac{π}{2}}{1 + c o s \frac{π}{2} + i s i n \frac{π}{2}})}^{2}

If $x_{n}$ = cos( $\frac{π}{4^{n}}$ ) + i sin( $\frac{π}{4^{n}}$ ) then $x_{1} x_{2} x_{3} . . . . . . . . . \infty =$

The polar form of complex number √3 + i

$\frac{1 - i}{1 + i}$ is equal to

Join BYJU'S Learning Program

Explore more

Geometrical Representation of Argument and Modulus

Standard XII Mathematics

Join BYJU'S Learning Program

The continued product of the four values of [cos(π3)+isin(π3)]3/4 is

The correct option is B 1We have, [cos(π3)+isin(π3)]3/4 =(cosπ+isinπ)1/4 Let z=(cosπ+isinπ)1/4 ∴z4=cosπ+isinπ=−1 If z1,z2,z3,z4 are the roots then z1z2z3z4=−1(−1)4−1=1

The continued product of the four values of ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$ is