If z r -cos-2 r - i sin-2 r where i -1- Find the value of z 1- z 2- z 3- z 4-

If nbsp;z_r = cos ( π/2^r ) + i sin ( π/2^r ) ................. (1) find the values of nbsp;z_1.z_2.z_3.z_4……∞ nbsp; by putting r = 1,2,3 ...... in equati ...

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Byju's Answer

Standard XI

Mathematics

General Solutions of (sin theta)^2 = (sin alpha)^2 , (cos theta)^2 = (cos alpha)^2 , (tan theta)^2 = (tan alpha)^2

If z r =cosπ/...

Question

If $z_{r}$ = cos $(\frac{π}{2^{r}})$ + i sin $(\frac{π}{2^{r}})$ where i = $\sqrt{- 1}$ . Find the value of $z_{1}$ . $z_{2}$ . $z_{3}$ . $z_{4} \dots \dots \infty$

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Solution

If $z_{r}$ = cos $(\frac{π}{2^{r}})$ + i sin $(\frac{π}{2^{r}})$ ................. (1)

find the values of $z_{1}$ . $z_{2}$ . $z_{3}$ . $z_{4} \dots \dots \infty$ by putting r = 1,2,3 ...... in equation (1)

$z_{1}$ . $z_{2}$ . $z_{3}$ . $z_{4} \dots \dots \infty$ ..............(2)

Substitute the value of $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4} \dots \dots \infty$ in equation 2.

= $[c o s \frac{π}{2} + i s i n \frac{π}{2}] [c o s \frac{π}{2^{2}} + i s i n \frac{π}{2^{2}}] [c o s \frac{π}{2^{3}} + i s i n \frac{π}{2^{3}}]$ ......

= $[c o s (\frac{π}{2} + \frac{π}{2^{2}} + \frac{π}{2^{3}} + \dots \dots \infty) + i s i n (\frac{π}{2} + \frac{π}{2^{2}} + \frac{π}{2^{3}} + \dots \dots \infty)]$

$\frac{π}{2}$ + $\frac{π}{2^{2}}$ + $\frac{π}{2^{3}} + \dots \dots \infty$ is a GP

Sum of infinite GP (when common ratio <1 ) = $\frac{a}{1 - r}$

So,

= cos $(\frac{\frac{π}{2}}{1 - \frac{1}{2}}) + i s i n (\frac{\frac{π}{2}}{1 - \frac{1}{2}})$

= $c o s (π) = i s i n (π) = - 1$

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If zr = cos (π2r) + i sin (π2r) where i = √−1 . Find the value of z1.z2.z3.z4……∞ __

If $z_{r}$ = cos $(\frac{π}{2^{r}})$ + i sin $(\frac{π}{2^{r}})$ where i = $\sqrt{- 1}$ . Find the value of $z_{1}$ . $z_{2}$ . $z_{3}$ . $z_{4} \dots \dots \infty$

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