Complex Numbers
Purely Imaginary Complex Expression — Trigonometric Equations
nta_pyq_2023_apr
Grade 11
Question:
Let $A=\left\{\theta\in(0,2\pi):\ \dfrac{1+2i\sin\theta}{1-i\sin\theta}\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is
$4\pi$
$3\pi$
$\pi$
$2\pi$
Step-by-Step Solution
Key Concept: Multiply numerator and denominator by conjugate, then set the real part equal to zero: $1-2\sin^2\theta=0\Rightarrow\cos 2\theta=0$.
Real part $=\frac{1-2\sin^2\theta}{1+\sin^2\theta}=0\Rightarrow\theta=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}$. Sum $=4\pi$.
Correct Answer: 1