Solving Equation for M & P - Geometry Explained

[yoyo]

Hi i am trying to solve this problem and i am stuck at finding all possible solutions, can someone please help?

Problem: Solve the following equation for M and P. Obtain all possible answers. Use the phasor method, and provide a geometrica diagram to explain the answers.

5*cos(Wo*t)=M*cos(Wo*t-(pi/6))+5*cos(Wo*t+p)
hint: Describe the figure in the z-plane given by the set {z:z=5*exp(j*p)-5 where 0<(or equal) p <(or equal) 2pi

when i converted to phasor form and solve for M, I got M=0, not sure what that means...

 

[Galileo]

Well, if you insert M=0 in the original equation, you can immediately see what p would be.

 

[quicknote]

It seems like you are on the right track with using the phasor method to solve this equation. To obtain all possible solutions, we need to consider the complex numbers involved in this problem. The equation can be rewritten as:

5*cos(Wo*t) = M*cos(Wo*t-(pi/6)) + 5*cos(Wo*t+p)

We can represent the complex numbers M and 5*exp(j*p) in the z-plane as points on the unit circle. The point representing M would be at an angle of -pi/6 from the positive real axis, while the point representing 5*exp(j*p) would be at an angle of p from the positive real axis.

The equation can then be visualized as a triangle in the z-plane, with one side representing 5*cos(Wo*t), another side representing M*cos(Wo*t-(pi/6)), and the third side representing 5*cos(Wo*t+p). The angle between the first two sides is -pi/6, and the angle between the second and third sides is p.

In order to solve for M, we need to find the length of the side representing M*cos(Wo*t-(pi/6)). Using the cosine law, we can write:

(M*cos(Wo*t-(pi/6)))^2 = (5*cos(Wo*t))^2 + (5*cos(Wo*t+p))^2 - 2*5*cos(Wo*t)*5*cos(Wo*t+p)*cos(-pi/6)

Simplifying this equation, we get:

(M*cos(Wo*t-(pi/6)))^2 = 25 + 25*cos(Wo*t+p) - 25*sin(Wo*t)*sin(p)

Since we are interested in finding all possible solutions, we can set this equation equal to zero and solve for M. This will give us two possible values for M, one when cos(Wo*t+p) = 1 and another when cos(Wo*t+p) = -1.

When cos(Wo*t+p) = 1, the equation simplifies to:

(M*cos(Wo*t-(pi/6)))^2 = 25

Solving for M, we get M=5 or M=-5. These values correspond to points on the unit circle that are at an angle of -pi/6 from the positive real axis, intersecting the circle at two points.

When cos(Wo