iScience said:Homework Statement
goal: solve for t; all else are constants
$$cos(\omega t)=1-e^{-(\frac{t}{RC})}$$
iScience said:Homework Statement
goal: solve for t; all else are constants
$$cos(\omega t)=1-e^{-(\frac{t}{RC})}$$Homework Equations
noneThe Attempt at a Solution
i turned the cos to complex notation & rearranged
$$e^{i\omega t}+e^{-(\frac{t}{RC})}=1$$
$$ln(e^{i\omega t}+e^{-(\frac{t}{RC})})=0$$
and i am stuck..
Complex numbers are numbers that contain both a real and an imaginary component. They are represented in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.
To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. For example, (2 + 3i) + (4 + 2i) = (2 + 4) + (3 + 2)i = 6 + 5i.
To multiply complex numbers, you can use the FOIL method just like with binomials. For example, (2 + 3i)(4 + 2i) = 2(4) + 2(2i) + 3i(4) + 3i(2i) = 8 + 4i + 12i + 6i^2 = -2 + 16i.
To divide complex numbers, you have to rationalize the denominator. This means multiplying the top and bottom by the conjugate of the denominator. For example, (4 + 2i) / (2 - 3i) = (4 + 2i)(2 + 3i) / (2 - 3i)(2 + 3i) = (8 + 12i + 4i + 6i^2) / (4 + 9i^2) = (2 + 16i) / 13 = 2/13 + (16/13)i.
Complex algebra is used in various fields of science and technology, such as engineering, physics, and computer science. It is also used in financial analysis and in creating mathematical models for real-life situations. For example, complex numbers are used in electrical engineering to represent the impedance of a circuit, and in computer graphics to represent 3D rotations and transformations.