If e=3 sin 3πt + 4 cos 3πt. what will be the rms value, peak value and phase of given emf?

1 Answer
Apr 6, 2018

The peak value is #5#

The RMS value is #5/sqrt2#

The phase angle is: #phi =-tan^-1(4/3)/(3pi)#

Explanation:

Using the identity

#sin(alpha+beta) = sin(alpha)cos(beta)+cos(alpha)sin(beta)#

We can write the following equation:

#e = Asin(3pit+C) = Asin(3pit)cos(C)+Acos(3pit)sin(C)#

Where A is the amplitude

Matching the given equation:

#e = 3sin(3pit) +4cos(3pit)#

We observe that:

#e= Asin(3pit+C)" [1]"#

#4=Asin(C)" [2]"#

#3 = Acos(C)" [3]"#

Divide equation [2] by equation [3]:

#tan(C) = 4/3#

#C = tan^-1(4/3)#

We can use equation [2] to solve for A:

#4 = Asin(tan^-1(4/3))#

We know that #sin(tan^-1(x)) = x/sqrt(1+x^2)#

#4 = A(4/3)/sqrt(1+(4/3)^2)#

#1 = A 1/sqrt(9+16)#

#A = 5#

An alternative form for #e# is:

#e = 5sin(3pit+ tan^-1(4/3))#

The peak value is #5#

The RMS value is #5/sqrt2#

We know that the phase angle of a sine wave of the form, #sin(Bt+C)#, is:

#phi = -C/B#

Matching this with the alternative form, the phase angle is

#phi =-tan^-1(4/3)/(3pi)#