please show how solve the system with help of tangent function?2x+(x^2)y=y 2y+(y^2)z=z 2z+(z^2)x=x

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You must figure out how to arrange each equation of the system suggesting the use of tangent function.

I propose you to write the top equation such as:

y-y(x^2)=2x (factorization is needed)

y(1-x^2)=2x (divide equation by 1-x^2)

y=2x/(1-x^2)

I propose you replacement of x=tan t:

y=2tan t/(1-(tan t)^2)

You are looking at the formula that expresses the tangent function of the angle 2t:y=tan(2t)

I propose you to write the middle equation such as:

z=2y/(1-y^2), where y=tan(2t)

z=2tan(2t)/(1-(tan(2t))^2)=tan2*(2t)=tan(4t)

I propose you to write the bottom equation such as:

x=2z/(1-z^2), where y=tan(4t)

x=2tan(4t)/(1-(tan(4t))^2)=tan2*(4t)=tan(8t)

use x=tant and x=tan(8t) and equate:

tan t=tan (8t) equivalent to t=8t+npi

t-8t=npi =>-7t=npi=>t=npi/-7

use t=-npi/7 to express x,y,z:

Answer: Solution of system: x=tan8npi/7 ; y= tan2npi/7 ; z=tan4npi/7.

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