RE: Solving a system of two trigonometric equations
May 20, 2025 at 6:11 pm
(This post was last modified: May 20, 2025 at 6:37 pm by arewethereyet.)
FlatAssembler posted a problem involving inverse trigonometric functions. He also posted the "alleged" solutions. The problem with forums like this, they are not conducive to precise mathematic notations and are sometimes are to interpret. But I think I have the problem presented more clearly.
1) atan(2/x) - atan(2/p) = -pi / 30
and
2) 2/p / 2/x = x / p =12.6
Also the the solution to the problem was supposed to be x = 0.2 and p = 0.016
In actuality the the solution isn't nice and neat and though there is an exact solution one needs to cut off to some acceptable decimal places I found x = 0.2286 and p = 0.01814
Its always difficult to describe a solution verbally but the problem isn't exactly so much a system of equations problem as it is a trigonometric problem, and going back to the basics. First one must recognize that statement 1) is the difference between two angles so atan(2/x) is an angle we'll call A and atan(2/p) is angle B So 1) becomes A - B = -pi/30
Then tan(A) = 2/x and tan(B) = 2/p thus if we draw two right triangles we have one with a height 2 and base x, and the other height 2 and base p but here's the trick p = x/a so tan(B)= 2a/x. Therefore we can draw two right triangles with the same base x and one with height 2 and the other with height 2 a.
We can then superimpose the two right triangles one over the other because the bases are both x. Now with this configuration an isosceles triangle is formed within whose sides are all determined via right triangle trig. Because we have three sides and three angles we can then apply the law of sins to the isosceles triangle. From the Law of Sines we can determine x because the problem reduces to a 4th order quadratic equation. We can reduce the 4th order variable using a substitution s = x^2. Then we solve for z using the quadratic formula and then solve for x once by taking the square root of z. If you are able to open up the five sheets in the google drive link I shared, I have provided a step by step solution. Its a pretty cool problem. Now I haven't seen a problem quite like this one before and I found it satisfying to solve.
The step by step solution is uploaded on the Google Drive and put the link to the drive below. There are five jpgs each numbered in order at the top right corner. Hopefully anyone interested can access the five sheets.
Pete Jeuck
1) atan(2/x) - atan(2/p) = -pi / 30
and
2) 2/p / 2/x = x / p =12.6
Also the the solution to the problem was supposed to be x = 0.2 and p = 0.016
In actuality the the solution isn't nice and neat and though there is an exact solution one needs to cut off to some acceptable decimal places I found x = 0.2286 and p = 0.01814
Its always difficult to describe a solution verbally but the problem isn't exactly so much a system of equations problem as it is a trigonometric problem, and going back to the basics. First one must recognize that statement 1) is the difference between two angles so atan(2/x) is an angle we'll call A and atan(2/p) is angle B So 1) becomes A - B = -pi/30
Then tan(A) = 2/x and tan(B) = 2/p thus if we draw two right triangles we have one with a height 2 and base x, and the other height 2 and base p but here's the trick p = x/a so tan(B)= 2a/x. Therefore we can draw two right triangles with the same base x and one with height 2 and the other with height 2 a.
We can then superimpose the two right triangles one over the other because the bases are both x. Now with this configuration an isosceles triangle is formed within whose sides are all determined via right triangle trig. Because we have three sides and three angles we can then apply the law of sins to the isosceles triangle. From the Law of Sines we can determine x because the problem reduces to a 4th order quadratic equation. We can reduce the 4th order variable using a substitution s = x^2. Then we solve for z using the quadratic formula and then solve for x once by taking the square root of z. If you are able to open up the five sheets in the google drive link I shared, I have provided a step by step solution. Its a pretty cool problem. Now I haven't seen a problem quite like this one before and I found it satisfying to solve.
The step by step solution is uploaded on the Google Drive and put the link to the drive below. There are five jpgs each numbered in order at the top right corner. Hopefully anyone interested can access the five sheets.
Pete Jeuck
Administrator Notice
Link removed per 30/30 rule. This is your last freebie. My suggestion is to read our rules.
Link removed per 30/30 rule. This is your last freebie. My suggestion is to read our rules.
