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jaylwood
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r(cos u + i sin u)
t(cos v + i sin v)
How do I convert these into exponential form using Euler's Theorem?
t(cos v + i sin v)
How do I convert these into exponential form using Euler's Theorem?
jaylwood said:r(cos u + i sin u)
t(cos v + i sin v)
How do I convert these into exponential form using Euler's Theorem?
jaylwood said:okay here is the problem i have. Given x = r(cos u + i sin u) and y = t(cos v + i sin v)
Prove that the amplitude of (xy) is the sum of their amplitudes. I don't understand where to go with it.
jaylwood said:rt eiu eiv What do i do to simplify that?
jaylwood said:ei(u+v)
jaylwood said:u+v ?
jaylwood said:what happens to the rt?
jaylwood said:So what would be my final answer?
… so the answer is that the amplitude of their sum is u + v, which is the sum of their amplitudes!jaylwood said:Prove that the amplitude of (xy) is the sum of their amplitudes.
Euler's Theorem is a mathematical formula that relates the trigonometric functions (sine, cosine, tangent, etc.) to the complex exponential function. It states that any complex number can be represented in the form of e^(ix), where e is the base of the natural logarithm, i is the imaginary unit, and x is the angle in radians.
Euler's Theorem has numerous applications in mathematics and physics. It allows us to simplify complex mathematical expressions and solve problems involving trigonometric functions. It also has connections to other important mathematical concepts, such as Fourier series and differential equations.
To convert from trigonometric to exponential form, we use the formula e^(ix) = cos(x) + i sin(x), where x is the angle in radians. This formula allows us to express any trigonometric function in terms of the complex exponential function.
Exponential form is often preferred over trigonometric form because it simplifies complex mathematical expressions and makes them easier to manipulate. It also allows us to perform operations such as differentiation and integration more easily.
Yes, Euler's Theorem can be extended to other types of functions, such as hyperbolic functions. This is known as Euler's formula and it states that e^(ix) = cos(x) + i sin(x) can be written as e^(x) = cosh(x) + i sinh(x), where x is a real number.