Question

What is $cis\;0$?

Answer 1

Hint: Euler’s law provides us that “any real number \[x\], ${e^{ix}} = \cos x + i\sin x$”
Where, \[e\]= base of natural logarithm
\[i\]= imaginary unit
\[x\]= angle in radians
This complex exponential function is sometimes expressed as $cis\;x$(“cosine plus I sine”). If \[x\]is a complex number, the formula still remains valid.
We can use trigonometric values to express the real and imaginary portions of an associated complex number.
In the standard (rectangular) form, a complex number would be expressed$a + ib$. However, on a complex number plane, the 'a' (real value) is corresponding with the x-axis and the 'b' (imaginary value) is corresponding with the y-axis. Therefore, any complex number (expressed as a coordinate pair on the plane) can be identified by its distance from the origin, r, and its vector, or angle, θ, above the positive x-axis.
Essentially, the coordinates (a,b) which express a complex number, are further converted into a polar equivalent, (r,θ) .
In this way, all complex numbers can be expressed as:
\[a + bi = r \cdot cis(\theta )\] Where: \[a = r \cdot cos(\theta )\]and \[b = r \cdot sin(\theta )\]
Therefore, \[r = \sqrt {{a^2} + {b^2}} \] and \[\theta = arctan\left( {\dfrac{b}{a}} \right)\]

Complete step-by-step answer:
So, here we are going to use Euler’s formula:
${e^{ix}} = \cos x + i\sin x$
The LHS of a equation can be written as $cis\;x$.
So, $cis\;x = \cos x + i\sin x$
Now as per our question, we have to find the value of $cis\;0$.
So, here we have $x = 0$
By substituting the above value of \[x\]into the expression, we get:
$ \Rightarrow cis\;0 = \cos 0 + i\sin 0$
We are already aware that the value of $\cos 0 = 1$and $\sin 0 = 0$.
So, now substituting these values in the above equation, we get
$ \Rightarrow cis\;0 = 1 + i\left( 0 \right) = 1$
So, the value of $cis\;0$ is $1$.

Note: While using Euler’s formula be careful what to substitute in $cis$and always keep $\sin x$ as imaginary if you keep $\cos x$ you may lead to wrong answer.
In some of the problems you may need to just directly use ${e^{ix}}$instead of using $cis$.
In addition to its application as a fundamental mathematical result, Euler's formula has many other uses in the world of physics and engineering.