If Zr - cosr-10 - i sinr-10 - Then- find the value of Z1 - Z2 - Z3 - Z4

Z_1 = cosπ10 + i sinπ10=e^iπ/10. Z_2 = cos2π10 + i sin2π10=e^i2π/10. Z_3 = cos3π10 + i sin3π10=e^i3π/10. Z_4 = cos4π ...

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Byju's Answer

Standard XII

Mathematics

Complex Numbers

If Zr = c...

Question

If $Z_{r} = (cos \frac{r π}{10} + i sin \frac{r π}{10}) .$ Then, find the value of $Z_{1} \cdot Z_{2} \cdot Z_{3} \cdot Z_{4}$

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Solution

$Z_{1} = cos \frac{π}{10} + i sin \frac{π}{10} = e^{i π / 10} .$

$Z_{2} = cos \frac{2 π}{10} + i sin \frac{2 π}{10} = e^{i 2 π / 10} .$

$Z_{3} = cos \frac{3 π}{10} + i sin \frac{3 π}{10} = e^{i 3 π / 10} .$

$Z_{4} = cos \frac{4 π}{10} + i sin \frac{4 π}{10} = e^{i 4 π / 10} .$

So,

$Z_{1} \cdot Z_{2} \cdot Z_{3} \cdot Z_{4}$

$= e^{\frac{π}{10}} e^{\frac{2 π}{10}} e^{\frac{3 π}{10}} e^{\frac{4 π}{10}}$

$= e^{\frac{π}{10} \cdot (1 + 2 + 3 + 4)}$

$= e^{i π} = c o s π + i s i n π = - 1$

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If Zr=(cosrπ10+isinrπ10). Then, find the value of Z1⋅Z2⋅Z3⋅Z4

Z1=cosπ10+isinπ10=eiπ/10.Z2=cos2π10+isin2π10=ei2π/10.Z3=cos3π10+isin3π10=ei3π/10.Z4=cos4π10+isin4π10=ei4π/10.So, Z1⋅Z2⋅Z3⋅Z4=eπ10e2π10e3π10e4π10=eπ10⋅(1+2+3+4)=eiπ=cosπ+isinπ=−1

If $Z_{r} = (cos \frac{r π}{10} + i sin \frac{r π}{10}) .$ Then, find the value of $Z_{1} \cdot Z_{2} \cdot Z_{3} \cdot Z_{4}$