Statement I The amplitude of sin (π/5)+i(1- cos (π/5)) is (π/10). Statement II If complex number lies on the first quadrant, then arg (z)=θ.

Question

Statement I The amplitude of sin (π/5)+i(1- cos (π/5)) is (π/10). Statement II If complex number lies on the first quadrant, then arg (z)=θ. (A) Statement I is correct (B) Statement II is correct (C) Both are correct (D) Neither I nor II is correct

Tardigrade Admin · Accepted Answer

Here, r cos θ= sin ((π/5)) and r sin θ=1- cos (π/5) ⇒ r cos θ=2 sin (π/10) cos (π/10) and r sin θ=2 sin 2 (π/10) ⇒ tan θ= tan ((π/10)) ⇒ θ=(π/10) Statements I and II are true.

Q. Statement I The amplitude of $sin \frac{π}{5} + i (1 - cos \frac{π}{5})$ is $\frac{π}{10}$ .
Statement II If complex number lies on the first quadrant, then $ar g (z) = θ$ .

Statement $I$ is correct

Statement II is correct

Both are correct

Neither I nor II is correct

Here, $r cos θ = sin (\frac{π}{5})$ and $r sin θ = 1 - cos \frac{π}{5}$
$\Rightarrow r cos θ = 2 sin \frac{π}{10} cos \frac{π}{10}$ and $r sin θ = 2 sin^{2} \frac{π}{10}$
$\Rightarrow tan θ = tan (\frac{π}{10}) \Rightarrow θ = \frac{π}{10}$
Statements I and II are true.

Q. Statement I The amplitude of sin5π​+i(1−cos5π​) is 10π​. Statement II If complex number lies on the first quadrant, then arg(z)=θ.

Statement I is correct