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hkBattousai
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a.cos(wt) + b.sin(wt) = M.cos(wt + ϕ)
Can you give me M and ϕ in terms of a and b?
Can you give me M and ϕ in terms of a and b?
No.hkBattousai said:The final representation was something like
M = sqrt(a^2 + b^2)
and
ϕ = arctan(-b/a)
but I'm no sure.
Can anyone confirm it for me?
M.cos(wt + ϕ) = a.cos(wt) + b.sin(wt)
cos(wt + ϕ) = (a/M).cos(wt) + (b/M).sin(wt)...(I)
cos(wt + ϕ) = cos(wt).cos(ϕ) - sin(wt).sin(ϕ)...(II)
From (I) and (II),
cos(ϕ) = (a/M)
sin(ϕ) = -(b/M)
cos^2(ϕ) + sin^2(ϕ) = (a^2 + b^2)/(M^2) = 1
We assume that M is always positive and we keep any negativity in the phase angle ϕ,
M = sqrt(a^2 + b^2)
sin(ϕ)/cos(ϕ) = tan(ϕ) = -(b/M)/(a/M) = -b/a
tan(ϕ) = -b/a ==> ϕ = arctan(-b/a)
hkBattousai said:Code:M.cos(wt + ϕ) = a.cos(wt) + b.sin(wt) cos(wt + ϕ) = (a/M).cos(wt) + (b/M).sin(wt)...(I) cos(wt + ϕ) = cos(wt).cos(ϕ) - sin(wt).sin(ϕ)...(II) From (I) and (II), cos(ϕ) = (a/M) sin(ϕ) = -(b/M) cos^2(ϕ) + sin^2(ϕ) = (a^2 + b^2)/(M^2) = 1 We assume that M is always positive and we keep any negativity in the phase angle ϕ, M = sqrt(a^2 + b^2) sin(ϕ)/cos(ϕ) = tan(ϕ) = -(b/M)/(a/M) = -b/a tan(ϕ) = -b/a ==> ϕ = arctan(-b/a)
Is there anything wrong in my derivation?
hkBattousai said:I liked your way of "you must do it yourself if you want to success"... :)
Landau said:You could also do the following:
t=0 gives a=M cos(ϕ),
t=pi/(2w) gives b=-M sin(ϕ).
Hence
a^2+b^2=M^2,
tan(ϕ)=-b/a.
The sum of cosine and sine can be represented as a single cosine expression by using the trigonometric identity: cos(x+y) = cos(x)cos(y) - sin(x)sin(y). This allows us to rewrite the sum of cosine and sine as a single cosine function, which simplifies the expression.
The purpose of representing the sum of cosine and sine as a single cosine expression is to simplify mathematical calculations and make them more efficient. This also allows us to easily identify patterns and relationships between different trigonometric functions.
No, the sum of cosine and sine can also be represented as a single sine expression by using the trigonometric identity: sin(x+y) = sin(x)cos(y) + cos(x)sin(y). However, representing it as a single cosine expression is often more useful in mathematical calculations.
Yes, there are limitations to representing the sum of cosine and sine as a single cosine expression. This method can only be used if the two functions have the same amplitude and frequency. If this is not the case, then the sum cannot be simplified into a single cosine expression.
In real-life applications, the sum of cosine and sine is often represented in terms of amplitude and phase. This allows us to model wave-like phenomena such as sound and electromagnetic waves. In these cases, the sum of cosine and sine is not simplified into a single cosine expression, but rather represented using a combination of trigonometric functions.