simplyfying a single single sinusoid

[mkforlife10]

simplyfying a single sinusoid

the question ask to simplify the following in a single sinsuoid x(t)= A+Bcos(ωt+θ)

here i the question
x(t)=5+5cos(200∏t+∏/2)+3cos(200∏t-∏/3)-2cos(200∏t-∏/4)

I am attaching what I did but not sure if it correct.
Since the ω∏t is same for all, I used euler form to convert the signal and then to rectangular form to add the real value and imaginary value and then back to Euler form and then in single sinusoid form.
euler formula is re^jθ=rcosθ+jrsinθ=rcos(ω+θ)

Thanks for the help

 

[ethelknox]

Determine the phase shift between the cosine function and the sine function. Use the trigonometry identity cos(x) = sin(x+Pi/2) to show that we can obtain the cosine function by shifting the sine wave Pi/2 to the left. The cosine function is therefore the sine function with a phase shift of -Pi/2.

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[MrAl]

the question ask to simplify the following in a single sinsuoid x(t)= A+Bcos(ωt+θ)

here i the question
x(t)=5+5cos(200∏t+∏/2)+3cos(200∏t-∏/3)-2cos(200∏t-∏/4)

I am attaching what I did but not sure if it correct.
Since the ω∏t is same for all, I used euler form to convert the signal and then to rectangular form to add the real value and imaginary value and then back to Euler form and then in single sinusoid form.
euler formula is re^jθ=rcosθ+jrsinθ=rcos(ω+θ)

Thanks for the help

Hello,

So where is your result, what did you get for the final answer?

You can also use the identities:

B*cos(wt+b)=B*(cos(b)*cos(wt)-sin(b)*sin(wt)), and
B*cos(wt-b)=B*(cos(b)*cos(wt)+sin(b)*sin(wt))

and

A*sin(wt)+B*cos(wt)=sqrt(A^2+B^2)*sin(wt+Ph)

where

Ph=atan2(B,A)