Can anyone please comment on this problem and advise me how to solve it?

# How can I get analytical solution of trigonometric equations?

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the constants are:

k11 = (16*V1*V1)/(n^3*(pi)^2*(2*pi*f)*L)

k22 = (16*V2*V2)/(n^3*(pi)^2*(2*pi*f)*L)

k33 = (16*V3*V3)/(n^3*(pi)^2*(2*pi*f)*L)

k12 = (8*V1*V2)/(n^3*(pi)^2*(2*pi*f)*L)

k13 = (8*V1*V3)/(n^3*(pi)^2*(2*pi*f)*L)

k23 = (8*V2*V3)/(n^3*(pi)^2*(2*pi*f)*L)

The equations are:

P1 = (k12.*cos(x(1)*pi/360).*cos(x(2)*pi/360).*sin(x(4)*pi/180))+(k13.*cos(x(1)*pi/360).*cos(x(3)*pi/360).*sin(x(5)*pi/180))

P2 = -(k12.*cos(x(1)*pi/360).*cos(x(2)*pi/360).*sin(x(4)*pi/180))+(k23.*cos(x(2)*pi/360).*cos(x(3)*pi/360).*sin((x(5)-x(4))*pi/180))

P3 = -(k13.*cos(x(1)*pi/360).*cos(x(3)*pi/360).*sin(x(5)*pi/180))+(k23.*cos(x(2)*pi/360).*cos(x(3)*pi/360).*sin((x(4)-x(5))*pi/180))

Q1 = (k11.*cos(x(1)*pi/360).*cos(x(1)*pi/360))-(k12.*cos(x(1)*pi/360).*cos(x(2)*pi/360).*cos(x(4)*pi/180))-(k13.*cos(x(1)*pi/360).*cos(x(3)*pi/360).*cos(x(5)*pi/180))

Q2 = -(k12.*cos(x(1)*pi/360).*cos(x(2)*pi/360).*cos(x(4)*pi/180))+(k22.*cos(x(2)*pi/360).*cos(x(2)*pi/360))-(k23.*cos(x(2)*pi/360).*cos(x(3)*pi/360).*cos((x(5)-x(4))*pi/180))

Q3 = -(k13.*cos(x(1)*pi/360).*cos(x(3)*pi/360).*cos(x(5)*pi/180))-(k23.*cos(x(2)*pi/360).*cos(x(3)*pi/360).*cos((x(5)-x(4))*pi/180))+(k33.*cos(x(3)*pi/360).*cos(x(3)*pi/360))

How can I solve for the angles x(1), x(2), x(3), x(4) and x(5)? Can anyone please help me to solve these equations?

### Accepted Answer

Walter Roberson
on 22 Jun 2018

Analytic solution:

x(1) = 180 + 360*Z1

x(2) = 180 + 360*Z2;

x(3) = 180 + 360*Z3;

x(4) and x(5) arbitrary (that is, the above 3 together solve all 5 equations)

Here, Z1, Z2, and Z3 represent arbitrary integers

##### 6 Comments

Walter Roberson
on 28 Jun 2018

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