If \(x+iy=\dfrac{1}{(1+cosθ)+isinθ}\),then the value of \(x^2+1\) is ?
\(\dfrac{7}{4}\)
\(\dfrac{13}{4}\)
\(\dfrac{1}{4}\)
\(\dfrac{9}{4}\)
\(\dfrac{5}{4}\)
The Correct Option is
Solution and Explanation
Given that:
\(x+iy=1/(1+cosθ)+isinθ\)
\(=\dfrac{1}{1+2cos^2(θ/2)+i.2sin(θ/2).cos(θ/2)}\)
\(=\dfrac{1}{1+2cos^2(θ/2)}.\dfrac{1}{e^{i.(θ/2)}}\)
\(=\dfrac{1}{1+2cos^2(θ/2)}.e^{-i.(θ/2)}\)
\(=\dfrac{1}{1+2cos^2(θ/2)}.(cos(θ/2)-isin(θ/2))\)
\(=\dfrac{1}{2}-itan(θ/2)\)
Hence, now we can write that \(x=\dfrac{1}{2}\)
So, \(x^2+1=(\dfrac{1}{2})^2+1=\dfrac{5}{4}\) (_Ans.)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.