Solving a Trigonometric System with Tangents

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[pkmath]

This seems like the problem I bumped into in AIME 2020 (slightly different in basic setup). 1st method looks like most of the students would try (as converting to algebraic form must be easier for more students) Very thorough explanation Sybermath!

[titassamanta]

Lwt tanx = p and tany = q. We have p+q=5 and 1/p + 1/q = 5/6 => (p+q)/pq = 5/6 => 5/pq=5/6 => pq = 6 => p(5-p)=6 [from (i), q=5-p] => p^2-5p+6=0. Hence p=3 or p=2. Thus we have two ordered pairs (tanx, tany) i.e. (3, 2) and (2, 3). Thus we have infinite solutions if our domain is all reals

[shmuelzehavi]

The first method is more simple and elegant than the second one. The second method is more complicated and incomplete:
The relation: tan(x)/tan(y) = 3/2 does not necessarily imply: tan(x) = 3 and tan(y) = 2.

[victorabaderamos]

Great video as always! I'm so glad I subscribed to this channel! Speaking of trigonometry, may I suggest you do this non-standard trig equation for the nex video?: xlog(sin(x))=10log_x cos(x), where the first log is the base 10 logarithm, and the second is base x log

[AbouTaim-Lille]

Putting tgx --x, tgy-->y we have: x+y =5, 1/x + 1/y = 5/6 --> x+y/x.y = 5/6
So 5 = x.y. 5/6, x.y=6 ==> x, y = 2, 3 and the solutions are arctg 2 and arctg 3

[udic]

The 2nd method is just solving the 1:23 set of equations in a different way.
3:42 i think it is better to write the y solution with mπ instead of nπ (to differentiate from the nπ of the x solution)

[MichaelRothwell]

Nice video as always. I used the first method.
There is a small error in the answer, which should be something like:
x=tan⁻¹2+nπ, y=tan⁻¹3+mπ, m, n ∈ Z
or
x=tan⁻¹3+nπ, y=tan⁻¹2+mπ, m, n ∈ Z
the point being that the multiples of π chosen for x and y are independent.

[jofx]

Title: with Tangent
My brain also thinking other way with cotangents

[kinshuksinghania]

By observation - tan x = 2, tan y = 3 or tan x = 3 and tan y = 2

[sattimama]

Tanx+tany=5, - 4*tanx*tany = 5*5-4*6=1--> tanx-tany=+-1, There are two set of solutions tanx=3, tany=2 or tanx=2, tany=3 --> x=arc
tan(3), y=arctan(2) or x=arctan(2), y=arctan(3).

[ipnorospo]

2nd one is the "rightest" method

[MrLidless]

A fancy version of X+Y = 5, XY = 6, only in this case there are an infinite number of solutions.

[klementhajrullaj]

Why tanx=2 and tany=3 to the second way, when tanv/tany=2/3=4/6=6/9 and so on?! ...

[theartist]

sorry but can I ask one question? can u explain me at 2:11

[larryfitzgerald]

Second denominator is tan x and not tan y

[scottleung]

I got everything except for the n*pi part.