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Expert: - 1/29/2009


Question
how do i express

i = 100sin(θ) + 150sin(θ + Π/3)

as Rsin(θ ± α) in radians

and find its maximum value

I have been trying to use the addition formula of
Rsin(A+B) = sinAcosB + cosAsinB
replacing the A and B with alpha and theta but there is no cos function in the question to use the formula

Answer

Phase Angle
Questioner:   Craig
Category:  Advanced Math
Private:  No
 
Subject:  Identities
Question:  how do I express

= 100sin(θ) + 150sin(θ + Π/3)

as R sin(θ ± α) in radians

and find its maximum value

I have been trying to use the addition formula of
Rsin(A+B) = sinAcosB + cosAsinB
replacing the A and B with alpha and theta but there is no cos function in the question to use the formula
...................................
Hi, Craig,

Something like:

p sin x + q sin(x + a)

will always be of the form:

r sin(x + h),  where  h  is called the phase angle.  If you plot your function, you will observe that it is just a sine curve.  So all you have to do (all, he says!) is write:

2 sin t + 3 sin(t + pi/3) =  R sin(t + h)   << took out the factor of 50
and solve:

2 sin t + 3 (sin t cos pi/3 + cos t sin pi/3) =  R sin(t + h)  << expand

2 sin t + 3 (sin t (1/2) + cos t (sq3/2) =  R sin(t + h)   << sq3 means sqrt(3)

2 sin t + 3 sin t (1/2) + 3 cos t (sq3/2) =  R sin(t + h)

7/2 sin t + 3 sq3/2 cos t = R sin(t + h)

7/2 sin t + 3 sq3/2 cos t = R (sin t cos h + cos t sin h)

7/2 sin t + 3 sq3/2 cos t = R sin t cos h + R cos t sin h

R cos h = 7/2  and  R sin h = 3 sq3 /2

Now square both and find that:

R^2 = (7/2)^2 + (3sq3/2)^2  << you figure out why this is OK.

R^2 = 49/4 + 27/4

R^2 = 76/4

R = sq76/2

Now divide both and find that:

tan h = 3sq3/7, and now solve.

This is a standard 'electrical phase angle' calculation.

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    Commentyes thanks for that, the question is part of an engineering course and it says it is used in electrical engineering so you were a lot of help.


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Paul Klarreich

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I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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