Euler's formula & expressing combinations of sin and cos as cos

[skweiler]

I have a rather difficult math problem. In my circuits class the professor prefers to express the cosine function using Euler's formula as (e^jwt+e^jwt)/2. Last semester for the final, one of the problem's solutions (that he gave us) used partial fraction decomposition to solve the Laplace transform of a circuit. I am certain that the math is correct but I do not understand how he came up with 22.36cos(0.5t+63.43) from V_c={(10(s-1)/[(s+j0.5)(s-j0.5)]} volts. After the partial fraction decomposition he comes up with A=10(1.118 at an angle of (-153.43+90) degrees) and B=10(1.118 at an angle of (153.43-90) degrees). This comes to 11.18 at an angle of +/-63.43 degrees. Translated into exponential form: 11.18[e^-j0.5t-63.43 +e^j0.5t+63.43]. What I don't understand is how he is able without a very complicated math formula (which I obtained from my former calculus professor) to go from this latest step to the answer. Will the argument of the cosine always be a positive constant multiplied by t with a positive phase angle? Note: in circuits the imaginary "i" is written as "j" to avoid confusion with current (i). Also, "w" is equal to lowercase omega. My goal in understanding this problem is to be able to express all sines and cosines as cosines and rid myself of the need to memorize the formula for converting the Laplace transform to sines and cosines.

 

[LCKurtz]

I have a rather difficult math problem. In my circuits class the professor prefers to express the cosine function using Euler's formula as (e^jwt+e^jwt)/2. Last semester for the final, one of the problem's solutions (that he gave us) used partial fraction decomposition to solve the Laplace transform of a circuit. I am certain that the math is correct but I do not understand how he came up with 22.36cos(0.5t+63.43) from V_c={(10(s-1)/[(s+j0.5)(s-j0.5)]} volts. After the partial fraction decomposition he comes up with A=10(1.118 at an angle of (-153.43+90) degrees) and B=10(1.118 at an angle of (153.43-90) degrees). This comes to 11.18 at an angle of +/-63.43 degrees. Translated into exponential form: 11.18[e^-j0.5t-63.43 +e^j0.5t+63.43].

I think you mean to have parentheses in those exponents:

11.18[e^{-j(0.5t-63.43)} +e^{j(0.5t+63.43)}]

What I don't understand is how he is able without a very complicated math formula (which I obtained from my former calculus professor) to go from this latest step to the answer. Will the argument of the cosine always be a positive constant multiplied by t with a positive phase angle? Note: in circuits the imaginary "i" is written as "j" to avoid confusion with current (i). Also, "w" is equal to lowercase omega. My goal in understanding this problem is to be able to express all sines and cosines as cosines and rid myself of the need to memorize the formula for converting the Laplace transform to sines and cosines.

He is using the Euler identities. You have

e^iθ = cos(θ) + i sin(θ)
e^-iθ = cos(θ) - i sin(θ)

Add these together

e^iθ + e^-iθ = 2 cos(θ)

Multiply by r and you have the identity he is using:

r(e^iθ + e^-iθ) = 2rcos(θ)

 

[skweiler]

Thank you this helps a lot.