# A Gorgeous Solution via Trig Substitution (WMC 70)

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**Published on Dec 13, 2018**- Let's solve our system of equations using familiar trigonometric identities.

Congratulations to PRAKHAR AGARWAL, Nicholas Patel, Hiren Bavaskar, santosh tripathy, fmakofmako, Nachiket Bhagade, Mr L, and Allaizn for successfully solving this math challenge question! PRAKHAR AGARWAL was the first person to solve the question.

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Welcome, everyone! My channel hosts one weekly math challenge question per week (made by either myself, my family, or my friends), which will be posted every Wednesday. Please comment your proposed answer and explanation below! If you are among the first ten people with the correct answer, you will be recognized in the next math challenge video. The solution to this question and new question will be posted next Wednesday.

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ThatOneGuy2 months agoi solved this without too many trig subtitution by turning the first equation into circle equation (turning sin^2 + cos^2 into y^2 + x^2 = 1), then solve for x/y in the 2nd eq and sub the value back to 1st equation

Math Zone3 months agoI love your video usclip.net/channel/UCLFe_L2xHeL0JQlUi9MJ15g?view_as=subscriber thanks!

JidtapadTK3 months agoI scream extremely loud in my mind when I see you sub

x = sin α

Love you 🥰

Chetra Kea6 months agoTeacher please write up cause down in close by writing load

el tapa6 months agoThis was delightful

GreenMeansGO7 months agoI hope people can read this and respond. I just solved this problem without even using the first equation(the one with two radicals). If we take

25(1-y^2)=41-40sqrt(1-x^2) and let x=cos(θ) and y=sin(θ) them the equation turns into

25(1-y^2)=41-40y implying that y=4/5 and x=3/5. Done.

GreenMeansGO7 months agoOh, wow. You are right. I got lucky. My solution assumes that (x,y) is a point on the unit circle which turned out to be correct in the end but was not necessarily true to begin with. Thank you for your feedback.

Ankush Agarwal7 months agoThis is not something you can do immediately.

Sinθ and cosθ are interrelated and substituting both of them in the equation for x and y is not right. In this question it just turned out to be correct. Thats not the case always as connection between x and y may not be the same as that of sinθ and cosθ.

Parth Chopra7 months agoThis was amazing!

Like next level clever...thanks mate!! :D

Shubham Saraf8 months agoWhat do you mean by exclusive?

Lumina Mathavan8 months agoGood mental exercise. Thank you sir

Jeffrey Cloete8 months agoThat was beautiful. .thanks!

human :38 months agoI can't help but notice That your "Hand"writing has become better over time; did you buy a drawing table?

Yerram Varun8 months agoThat was awesome! Love your channel

Rajendra Misir8 months agoTerrific job Prakhar Agarwal! I find it amazing that equation 1 can be rewritten as the equation of the unit circle using trigonometric identities such as the half angle of tangent function(tan(x/2).

gajra rangare8 months agoNice Problem.

Paul Murray8 months agoSeeing sqrt(1-x^2), I was thinking that the way to go would be to redo the whole thing in polar coordinates.

dunkelheit8 months agoWhy has he used two different angles? Is it wrong to use only one? And if it is wrong, why is it wrong? Thanks for your time.

dunkelheit7 months ago@π Super J π Thank you so much!

π Super J π8 months agoIf you used the same angle then you would be assuming x and y are the same value because sine is bijective between 0 and 90.

Anandan Poornash8 months ago^{+1}very nice solution

el tapa8 months agoAhhhh very nice. I didn't knew that tangent identity

יהודה שמחה ולדמן8 months ago^{+1}Did you get my solution in email?

LetsSolveMathProblems8 months ago^{+2}Only the solutions posted as USclip comments will be considered. =)

Shanmuga Sundaram8 months agoVery nice illustration.Thank you.

adandap8 months ago^{+12}The trig solution is lovely. I wouldn't have thought of it myself, but worked it through after noticing it in Prakhar's answer. That said, the algebraic approach is nicer than it looks at first glance because of the way that (1 - sqrt(1-x^2))/x rationalises and inverts. And the y term simplifies nicely to Sqrt( (1+y)/(1-y)).

Slight Lokii8 months ago^{+3}What is your reasoning that alpha and beta are complimentary?

Sea cucumber8 months ago@Slight Lokii Happy to help :)

Slight Lokii8 months ago^{+1}Sea cucumber oh I understand now! Thanks :)

Sea cucumber8 months ago^{+5}I'm going to use a for alpha and b for beta:

tan a/2 = tan (45 - b/2)

The following is a well known tangent identity:

If tan x = tan y then x=y + k×360° where k is some whole number. Therefore:

a/2 = 45 - b/2 + k×360°

Rearranging gives you:

a+b=90 + k×360°

However, we set both a and b to be between 0° and 90° so we cannot add or subtract any whole number multiple of 360° except for 0. So:

a+b=90° + 0×360°

a+b=90° which is the definition of complementary angles.

Mi Les8 months ago^{+3}“gorgeous”

It’s just too much algebra just to get nice answer lol

Antonio Banda8 months ago^{+5}Face reveal please

santhanam krishnan bhaskar2 months agosearch up blackpenredpen

Roushd Hassam7 months agowhy

Cristhian Martínez8 months ago^{+1}Amazing

Ben Burdick8 months ago^{+7}I feel really dumb for not seeing the trig sub when I attempted this problem.

Santhosh Kumar8 months ago^{+7}Thank you very much for the solution. I am big fan of your channel.

Santhosh Kumar8 months agoSir,

Thank you for the heart . Don't stop uploading difficult problems with simple solutions. I am eagerly waiting for the next weekly challenge number 71 .

Pete Berg8 months agoIn what cases again i can use trig substitution?

What if the question doesnt tell that 0

Jerry Tan8 months agoJust want to add a complement on Prathmesh's answer. In this specific question, notice that abs(x) and abs(y) has to be between 0 and 1 as sqrt(1-x^2) and sqrt(1-y^2) are both undefined when abs(x) or abs(y) is greater than 1 (you can't have a negative inside the square root for Real Numbers). Therefore, in this specific question, even if the information (0

Pete Berg8 months ago^{+1}@Prathmesh Joshi thx. That might be a clue

Prathmesh Joshi8 months ago^{+1}note that -1

Pete Berg8 months agoWell, this open my mind. I hope i can apply this strategy as my tool

Ducksfan1018 months ago^{+54}Okay that’s epic.

bridogg1548 months agoSolve this one usclip.net/video/m0PqvrZbcTo/video.html