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Question: solving intersection points of two function.

Question:solving intersection points of two function.

The function 110 Maple
solve rootfinding fsolve trig
September 30 2021
2

Hello! 

I dont like cursing, but this had got me going. 

How are there no solutions for the intersection points, i am only getting some, but not all of them, but i did ask Maple to produce an answer, but it however failed to produce all the answers. Its not that i did not try different techniques. 

I really wonder what went wrong. How could this be?! Just Maple... Answers between zero and 2 Pi... plz.... (!!!)

The answers i get with the suggested method from my math book dont give any straight answer. How is that an answer, i dont want imaginary numbers, i dont want to have  Z values (what is with that anyways?)..

Well here is what i found. Also if i were to ask for solutions between x values closer together i got more answers that were not there before. That is not what i want, that is like manual labor, that is not what i want at all, i want to be faster not slower. My graphing calculator was even a faster and that did get all the answers right away.. 

Any way, it missed an answer between 1/2Pi and Pi, answers that were not there before when looking for values between 0 and 2Pi. 

So, so odd..


It says in english: Also with finding solutions for solving goniometric equations with Maple, we should not forget that there are more solutions than just one, most of the times. We give you two methods to find all the solutions for solving goniometric equations. 

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#first attempt
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evalf(solve({x > 0*Pi, sin(x) = cos(4*x-(1/6)*Pi), x < 2*Pi}, x, allsolutions, explicit)); smartplot(sin(x), cos(4*x-(1/6)*Pi)); plot([sin(x), cos(4*x-(1/6)*Pi)], x = 0 .. 2*Pi, y = 0 .. 1); evalf(solve({x > (1/2)*Pi, sin(x) = cos(4*x-(1/6)*Pi), x < Pi}, x, allsolutions, explicit))

{x = 4.188790204}, {x = 5.445427267}

  

 

 

{x = 1.675516082}, {x = 2.932153141}

 (1)
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#second attempt
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"f(x):=sin(x); g(x):= cos(4 x-1/(6)*Pi);fsolve([f,g]);"

proc (x) options operator, arrow, function_assign; sin(x) end proc

  

proc (x) options operator, arrow, function_assign; cos(4*x-(1/6)*Pi) end proc

  

[6.8067840827778841, -9.1035796650957585]

 (2)
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#third attempt;
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_EnvAllSolutions := true; solve(sin(x) = cos(4*x-(1/6)*Pi), x)

-(2/3)*Pi+2*Pi*_Z4, -arctan((1/6)*3^(1/2)*(2*((1/2)*(4+(4*I)*3^(1/2))^(1/3)+2/(4+(4*I)*3^(1/2))^(1/3))^2-(1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)-4)/((1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)))+2*Pi*_Z5, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)+((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))-Pi+2*Pi*_Z5, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)-((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))+Pi+2*Pi*_Z5, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1))+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2))+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3))+Pi+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4))+Pi+2*Pi*_Z6

 (3)
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#fourth attempt;
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RootOf(sin(x) = cos(4*x-(1/6)*Pi), x); allvalues(%)

RootOf(sin(_Z)-sin(4*_Z+(1/3)*Pi))

  

-(2/3)*Pi+2*Pi*_Z7, -arctan((1/6)*3^(1/2)*(2*((1/2)*(4+(4*I)*3^(1/2))^(1/3)+2/(4+(4*I)*3^(1/2))^(1/3))^2-(1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)-4)/((1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)))+2*Pi*_Z8, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)+((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))-Pi+2*Pi*_Z8, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)-((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))+Pi+2*Pi*_Z8, arctan((1/6)*3^(1/2)*(2*(1/4+(1/4)*5^(1/2)+(1/4)*(30-6*5^(1/2))^(1/2))^3+1/4-(7/4)*5^(1/2)-(7/4)*(30-6*5^(1/2))^(1/2))/(1/8+(1/8)*5^(1/2)+(1/8)*(30-6*5^(1/2))^(1/2)))+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4+(1/4)*5^(1/2)-(1/4)*(30-6*5^(1/2))^(1/2))^3+1/4-(7/4)*5^(1/2)+(7/4)*(30-6*5^(1/2))^(1/2))/(1/8+(1/8)*5^(1/2)-(1/8)*(30-6*5^(1/2))^(1/2)))+Pi+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4-(1/4)*5^(1/2)+(1/4)*(30+6*5^(1/2))^(1/2))^3+1/4+(7/4)*5^(1/2)-(7/4)*(30+6*5^(1/2))^(1/2))/(1/8-(1/8)*5^(1/2)+(1/8)*(30+6*5^(1/2))^(1/2)))+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4-(1/4)*5^(1/2)-(1/4)*(30+6*5^(1/2))^(1/2))^3+1/4+(7/4)*5^(1/2)+(7/4)*(30+6*5^(1/2))^(1/2))/(1/8-(1/8)*5^(1/2)-(1/8)*(30+6*5^(1/2))^(1/2)))+Pi+2*Pi*_Z9

 (4)
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