# Express exp(3+Pi*i) in Cartesian Form

• nicemaths
So, in summary, to express exp(3+π*i) in Cartesian Form, we can equate e^(3+π*i) to e^(x)e^(iy) and solve for x and y using the equations e^(x)cos(y) = 3 and e^(x)sin(y) = π. This gives us x=3 and y=π, which means that exp(3+π*i) in Cartesian Form is e^3(cosπ + isinπ).
nicemaths
The problem statement
Express exp(3+π*i) in Cartesian Form.

The attempt at a solution
Equating
e^(3+πi) = e^(x)e^(iy) = e^(x)(cos(y) + isin(y))
then
e^(x)cos(y) = 3
e^(x)sin(y)=π
now
|e^(3+πi)| = e^(x)
so x = sqrt(9+π^2)
then
cos(y) = 3/sqrt(9+π^2)
sin(y) = π/sqrt(9+π^2)

at this point i don't know where to go to find y,
i did tan(y) = π/3 but after than i don't know where to gothanks for the help !

nicemaths said:
e^(x)cos(y) = 3
e^(x)sin(y)=π
How do you get this? x=3, y=π. What is e3? What is cos π? What is sin π?

mjc123 said:
How do you get this? x=3, y=π. What is e3? What is cos π? What is sin π?

oh deary me, i feel silly now that i see the huge error

No need to feel silly, we all make mistakes, just learn from them.

## 1. What is the value of exp(3+Pi*i)?

The value of exp(3+Pi*i) is approximately -20.0855 + 1.2246i in Cartesian form.

## 2. How do you express exp(3+Pi*i) in Cartesian form?

To express exp(3+Pi*i) in Cartesian form, use the formula: exp(3+Pi*i) = e^3 * (cos(Pi) + i*sin(Pi)). This simplifies to -20.0855 + 1.2246i.

## 3. What is the significance of exp(3+Pi*i)?

Exp(3+Pi*i) is a complex number that represents the exponential function with a complex argument. It is commonly used in mathematical and scientific calculations.

## 4. Can you explain the role of e and Pi in the expression exp(3+Pi*i)?

The number e is the base of the natural logarithm and Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter. In the expression exp(3+Pi*i), e is raised to the power of 3 and Pi*i represents a complex angle, which together form a complex number in Cartesian form.

## 5. How is exp(3+Pi*i) related to the polar form of a complex number?

The polar form of a complex number is expressed as r(cos(theta) + i*sin(theta)), where r is the magnitude of the complex number and theta is the angle in radians. In the case of exp(3+Pi*i), the magnitude is equal to e^3 and the angle is Pi, which is equivalent to 180 degrees. Therefore, the Cartesian form -20.0855 + 1.2246i is equivalent to the polar form of 20.2026(cos(180) + i*sin(180)).

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