# Complex number(exponential form)

• naspek
In summary, the conversation is about finding a possible value for w that satisfies the relation e^w = Z, where z = 1 + 3^1/2. The person provides their solution of w = i(pi/3) and asks for verification, which is confirmed by another person using a different method.
naspek

## Homework Statement

hey there.. I've try this question(below)..
but, i don't know whether my solution is correct or not..

Find a possible value for w that satisfies relation...
e^w = Z -->where z = 1 + 3^1/2

my solution...
r = 2
theta = pi/3
change z into exponential form...
z = 2e^i(pi/3)

hence, w = i(pi/3)

can anybody pleas guide me or verified my answer...

You mean
R.e^w = Z -->where z = 1 + 3^1/2.j

If r = 2
and w = j (pi/3)

Knowing
R . e^(j.t) = R. [cos(t) + j sin(t)],

2 . e^(j. pi /3) = ... ?

rootX said:
2 . e^(j. pi /3) = ... ?

2 . e^(j. pi /3) = 2[cos (pi/3) + i sin (pi/3)]

naspek said:
2 . e^(j. pi /3) = 2[cos (pi/3) + i sin (pi/3)]

Which equals z, hence verifying that your solution is good.

ok.. so.. is my answer is correct?

rootX said:
...verifying that your solution is good.
naspek said:
ok.. so.. is my answer is correct?
What rootX said is another way of saying "yes, it is correct"

thanks guys! really appreciate your help.. =)

## 1. What is a complex number in exponential form?

A complex number in exponential form is a mathematical expression of the form reix, where r is the modulus (or absolute value) of the complex number and x is the argument (or angle) of the complex number. It is often used to represent complex numbers in polar coordinates.

## 2. How is a complex number in exponential form different from a complex number in standard form?

A complex number in exponential form is a different way of expressing a complex number than the standard form, which is a + bi, where a and b are real numbers and i is the imaginary unit. While both forms represent the same complex number, exponential form allows for easier calculations involving complex numbers, especially when raising them to powers or taking roots.

## 3. What is the relationship between complex numbers in exponential form and trigonometric functions?

Complex numbers in exponential form have a close relationship with trigonometric functions, specifically sine and cosine. This is because eix = cos(x) + i sin(x), known as Euler's formula. This relationship is useful in solving problems involving complex numbers, as it allows us to use our knowledge of trigonometric functions to better understand and manipulate complex numbers.

## 4. How do you convert a complex number from exponential form to standard form?

To convert a complex number from exponential form to standard form, we use the trigonometric identity eix = cos(x) + i sin(x), where x is the argument (or angle) of the complex number. First, we find the cosine and sine values of x, and then we substitute them into the formula, replacing x with the argument of the complex number. Finally, we combine like terms to get the complex number in standard form.

## 5. How are complex numbers in exponential form used in real life?

Complex numbers in exponential form have many real-life applications, particularly in fields such as engineering, physics, and signal processing. They are used to represent alternating currents and in the study of electrical circuits. They are also used in modeling and predicting the behavior of waves, such as sound waves and electromagnetic waves. In addition, they are used in solving differential equations, which have numerous applications in science and engineering.

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