The complex number $z=\dfrac{i-1}{\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}}$ is equal to:
$\sqrt{2}\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)$
$\cos\dfrac{\pi}{12}-i\sin\dfrac{\pi}{12}$
$\sqrt{2}\left(\cos\dfrac{\pi}{12}+i\sin\dfrac{\pi}{12}\right)$
$\sqrt{2}\,i\left(\cos\dfrac{5\pi}{12}-i\sin\dfrac{5\pi}{12}\right)$
Step-by-Step Solution
Key Concept: $i-1=\sqrt{2}e^{i3\pi/4}$. Denominator $=e^{i\pi/3}$. $z=\sqrt{2}e^{i(3\pi/4-\pi/3)}=\sqrt{2}e^{i5\pi/12}$.
Step 1:
To find the value of the complex number $z$, we start by expressing the given complex number in terms of its components using Euler's formula, which states that $e^{i\theta} = \cos\theta + i\sin\theta$. The given complex number is $z=\dfrac{i-1}{\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}}$. We recognize that the denominator can be expressed as $e^{i\frac{\pi}{3}}$.
Step 2:
Using Euler's formula, we can rewrite the denominator as $\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3} = e^{i\frac{\pi}{3}}$. Therefore, the complex number $z$ becomes $z = \dfrac{i-1}{e^{i\frac{\pi}{3}}}$. To simplify this expression, we can multiply both the numerator and the denominator by $e^{-i\frac{\pi}{3}}$ to get rid of the complex denominator.
Step 3:
Multiplying by $e^{-i\frac{\pi}{3}}$, we have $z = \dfrac{(i-1)e^{-i\frac{\pi}{3}}}{e^{i\frac{\pi}{3}}e^{-i\frac{\pi}{3}}} = \dfrac{(i-1)e^{-i\frac{\pi}{3}}}{1} = (i-1)e^{-i\frac{\pi}{3}}$. Now, we express $i-1$ in polar form to simplify the multiplication.
Step 4:
The complex number $i-1$ can be written in polar form. Its magnitude is $\sqrt{1^2 + 1^2} = \sqrt{2}$, and its angle $\theta$ with the positive x-axis is $\tan^{-1}\left(\frac{1}{-1}\right) = \frac{3\pi}{4}$ because it lies in the second quadrant. Thus, $i-1 = \sqrt{2}e^{i\frac{3\pi}{4}}$. Substituting this back into our expression for $z$, we get $z = \sqrt{2}e^{i\frac{3\pi}{4}}e^{-i\frac{\pi}{3}}$.
Step 5:
Using the properties of exponents, we can combine the two exponential terms: $z = \sqrt{2}e^{i\left(\frac{3\pi}{4} - \frac{\pi}{3}\right)}$. Simplifying the angle, $\frac{3\pi}{4} - \frac{\pi}{3} = \frac{9\pi}{12} - \frac{4\pi}{12} = \frac{5\pi}{12}$. Therefore, $z = \sqrt{2}e^{i\frac{5\pi}{12}}$.
Step 6:
Finally, expressing $e^{i\frac{5\pi}{12}}$ in terms of sine and cosine using Euler's formula, we have $z = \sqrt{2}\left(\cos\dfrac{5\pi}{12} + i\sin\dfrac{5\pi}{12}\right)$. This matches Option 1. Therefore, the final answer is $\boxed{1}$, which corresponds to $\sqrt{2}\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)$.
Correct Answer: 1