The modulus- amplitude form of \(\sqrt 3 + i\) , where \(i = \sqrt { - 1}\) is
Understanding Modulus-Amplitude Form of Complex Numbers
The modulus-amplitude form, also known as the polar form, represents a complex number \(z = x + iy\) in terms of its distance from the origin (modulus, \(r\)) and the angle it makes with the positive x-axis (amplitude or argument, \(\theta\)). The general form is given by \(z = r(\cos \theta + i \sin \theta)\).Finding Modulus and Amplitude for \(\sqrt 3 + i\)
We are given the complex number \(z = \sqrt 3 + i\). This is in the Cartesian form \(x + iy\), where \(x = \sqrt 3\) and \(y = 1\).Calculating the Modulus
The modulus \(r\) of a complex number \(z = x + iy\) is calculated using the formula \(r = \sqrt{x^2 + y^2}\). For \(z = \sqrt 3 + i\): \(r = \sqrt{(\sqrt 3)^2 + (1)^2}\) \(r = \sqrt{3 + 1}\) \(r = \sqrt{4}\) \(r = 2\) So, the modulus of \(\sqrt 3 + i\) is 2.Finding the Argument (Amplitude)
The argument \(\theta\) is the angle such that \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\). Using the values \(x = \sqrt 3\), \(y = 1\), and \(r = 2\): \(\cos \theta = \frac{\sqrt 3}{2}\) \(\sin \theta = \frac{1}{2}\) Since both \(\cos \theta\) and \(\sin \theta\) are positive, the angle \(\theta\) lies in the first quadrant. The angle whose cosine is \(\frac{\sqrt 3}{2}\) and sine is \(\frac{1}{2}\) is \(\frac{\pi}{6}\) radians (or 30 degrees). Therefore, the argument \(\theta = \frac{\pi}{6}\).Writing the Modulus-Amplitude Form
Now, substituting the values of \(r\) and \(\theta\) into the modulus-amplitude form \(z = r(\cos \theta + i \sin \theta)\): \(z = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) This is the modulus-amplitude form of the complex number \(\sqrt 3 + i\).Comparing with Options
Let's compare our result \(2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) with the given options to find the correct representation. The calculated form matches one of the provided options.| Component | Value |
|---|---|
| Complex Number \(z\) | \(\sqrt 3 + i\) |
| Real Part \(x\) | \(\sqrt 3\) |
| Imaginary Part \(y\) | \(1\) |
| Modulus \(r\) | \(2\) |
| Argument \(\theta\) | \(\frac{\pi}{6}\) |
| Modulus-Amplitude Form | \(2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) |
The modulus-amplitude form is \(2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\).
Similar Questions
What is the value of \({\left[ {\frac{{i + \sqrt 3 }}{2}} \right]^{2019}} + {\left[ {\frac{{i - \sqrt 3 }}{2}} \right]^{2019}}?\)
What is the modulus of z?
What is the principal argument of z?
What is the value of \({\left( {\frac{{ - 1 + i\sqrt 3 }}{2}} \right)^{3n}} + {\left( {\frac{{ - 1 - i\sqrt 3 }}{2}} \right)^{3n}}\) ?
Where \(i = \sqrt { - 1} ?\)
Which one of the following is correct in respect of the cube roots of unity?
The number of non-zero integral solutions of the equation |1 - 2i| x= 5 xis
If α and β are different complex numbers with |α | = 1, then what is \(\left| {\frac{{\alpha - \beta }}{{1 - \alpha \bar \beta }}} \right|\) equal to?
What is the principal argument of (-1 –i), where i = \(\sqrt { - 1}\)
Let α and β be real numbers and z be a complex number. If z 2+ αz + β = 0 has two distinct non-real roots with Re(z) = 1, then it is necessary that.
What is i 1000 + i 1001 + i 1002 + i 1003 equal to (where i \(= \sqrt { - 1}\) )?
Important Questions from Complex Numbers
If A + iB = tan (x + iy), then the value of tan 2x is?
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
