Solveeit Logo
Login

Question

Question: If x<sub>r</sub> = cos \(\frac{\pi}{2^{r}}\) + i sin \(\frac{\pi}{2^{r}}\), z<sub>t</sub> = cos \(\f...

If xr = cos π2r\frac{\pi}{2^{r}} + i sin π2r\frac{\pi}{2^{r}}, zt = cos π3t\frac{\pi}{3^{t}} + i sin π3t\frac{\pi}{3^{t}} r = 1, 2, 3, …., t = 1, 2, 3, …. The value of (x1 x2 x3 ….. )2 (z1 z2 z3 …… )4 –

A

0

B

1

C

–1

D

i

Answer

1

Explanation

Solution

Sol. (x1 x2 x3 ….. )2 (z1 z2 z3 …. )4

[(cos⁡π2+isin⁡π2)(cos⁡π22+isin⁡π22)(cos⁡π23+isin⁡π23)....∞]2\left\lbrack \left( \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} \right)\left( \cos\frac{\pi}{2^{2}} + i\sin\frac{\pi}{2^{2}} \right)\left( \cos\frac{\pi}{2^{3}} + i\sin\frac{\pi}{2^{3}} \right)....\infty \right\rbrack^{2}

[(cos⁡π3+isin⁡π3)(cos⁡π32+isin⁡π32)+....∞]4\left\lbrack \left( \cos\frac{\pi}{3} + i\sin\frac{\pi}{3} \right)\left( \cos\frac{\pi}{3^{2}} + i\sin\frac{\pi}{3^{2}} \right) + ....\infty \right\rbrack^{4}

= [cos⁡(π3+π32+π33+...)+isin⁡(π3+π32+π33+....)]4\left\lbrack \cos\left( \frac{\pi}{3} + \frac{\pi}{3^{2}} + \frac{\pi}{3^{3}} + ... \right) + i\sin\left( \frac{\pi}{3} + \frac{\pi}{3^{2}} + \frac{\pi}{3^{3}} + .... \right) \right\rbrack^{4}

=[cos⁡(π/31−13)+isin⁡(π/31−13)]4\left\lbrack \cos\left( \frac{\pi/3}{1 - \frac{1}{3}} \right) + i\sin\left( \frac{\pi/3}{1 - \frac{1}{3}} \right) \right\rbrack^{4}

(cos p + i sin p)2 (cos p/2 + i sin p/2)4 = (–1)2 (i)4 = 1.