Proving Eiθ = cos θ + i sen θ: A Scientific Exploration

Thread starter eljota38
Start date Mar 16, 2013
Tags

Cos

In summary, the equation Eiθ = cos θ + i sen θ is a widely used equation in mathematics and physics, particularly in the fields of trigonometry, complex analysis, and quantum mechanics. It represents the relationship between the angle θ and the complex number Eiθ, and can be derived from Euler's formula. This equation can be used to simplify complex numbers and is closely related to the trigonometric unit circle.

Mar 16, 2013

eljota38

Can anyone prove me this stament please.

Physics news on Phys.org

Mar 16, 2013

gnrlies00

the proof to Euler's formula

e^iθ=cosθ+isinθ

is using taylor series for the exponential function

For any complex number z, we define e^z by

e^z=[itex]\sum[/itex][itex]\frac{z^n}{n!}[/itex].

Mar 16, 2013

eljota38

thanks a lot I think you were a great help

Mar 16, 2013

micromass

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Please read this: https://www.physicsforums.com/blog.php?b=3588

Last edited by a moderator: May 6, 2017

FAQ: Proving Eiθ = cos θ + i sen θ: A Scientific Exploration

What is the equation Eiθ = cos θ + i sen θ used for?

The equation Eiθ = cos θ + i sen θ is frequently used in mathematics and physics, particularly in the fields of trigonometry, complex analysis, and quantum mechanics. It is also commonly used in engineering and signal processing.

What does the variable θ represent in the equation Eiθ = cos θ + i sen θ?

The variable θ represents the angle in radians. In trigonometry, radians are used to measure angles based on the radius of a circle, where one full rotation is equal to 2π radians.

How is the equation Eiθ = cos θ + i sen θ derived?

The equation Eiθ = cos θ + i sen θ is derived from Euler's formula, which states that eix = cos x + i sin x. By substituting θ for x, we get Eiθ = cos θ + i sen θ, where E is the base of the natural logarithm.

Can the equation Eiθ = cos θ + i sen θ be used to simplify complex numbers?

Yes, the equation Eiθ = cos θ + i sen θ can be used to simplify complex numbers in the form of a + bi, where a and b are real numbers and i is the imaginary unit. This equation allows us to express complex numbers in terms of trigonometric functions, which can be useful in solving complex equations.

How is the equation Eiθ = cos θ + i sen θ related to the unit circle?

The equation Eiθ = cos θ + i sen θ is closely related to the unit circle in trigonometry. When we graph the real and imaginary parts of the equation, they form a circle with a radius of 1, known as the unit circle. This circle is used to visualize trigonometric functions and their relationships.