The general value of the real angle θ, which satisfies the equation, (cosθ + i sinθ) (cos2θ + i sin2θ)........ (cosnθ + i sinnθ) = 1 is given by, (assuming k is an integer)

Question

The general value of the real angle θ, which satisfies the equation, (cosθ + i sinθ) (cos2θ + i sin2θ)........ (cosnθ + i sinnθ) = 1 is given by, (assuming k is an integer) (A) (2kπ/n+2) (B) (4kπ/n(n+1)) (C) (4kπ/n+1) (D) (6kπ/n(n+1))

Tardigrade Admin · Accepted Answer

eiθ⋅ ei(2θ)⋅ ei(3θ) ldots ei(nθ) = 1 ⇒ ei(θ ((n(n+1))/2)) = ei2k π ⇒ θ = (4kπ/n(n+1))

Q. The general value of the real angle $θ$ , which satisfies the equation, (cos $θ$ + i sin $θ$ ) (cos2 $θ$ + i sin2 $θ$ )........ (cosn $θ$ + i sinn $θ$ ) = 1 is given by, (assuming k is an integer)

$\frac{2 kπ}{n + 2}$

$\frac{4 kπ}{n ( n + 1 )}$

$\frac{4 kπ}{n + 1}$

$\frac{6 kπ}{n ( n + 1 )}$

$e^{i θ} \cdot e^{i (2 θ)} \cdot e^{i (3 θ)} \dots e^{i (n θ)} = 1 \Rightarrow e^{i (θ \frac{( n ( n + 1 ) )}{2})} = e^{i 2 k π \Rightarrow θ = \frac{4 kπ}{n ( n + 1 )}}$

Q. The general value of the real angle θ, which satisfies the equation, (cosθ + i sinθ) (cos2θ + i sin2θ)........ (cosnθ + i sinnθ) = 1 is given by, (assuming k is an integer)

n+22kπ​

n(n+1)4kπ​

n+14kπ​

n(n+1)6kπ​