# 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.
Your support is truly a huge encouragement.
Every subscriber and every like are wholeheartedly appreciated.
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.
For more Weekly Math Challenges:

• ThatOneGuy 2 months ago

i 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 Zone 3 months ago

I love your video usclip.net/channel/UCLFe_L2xHeL0JQlUi9MJ15g?view_as=subscriber thanks!

I scream extremely loud in my mind when I see you sub
x = sin α
Love you 🥰

• Chetra Kea 6 months ago

• el tapa 6 months ago

This was delightful

• GreenMeansGO 7 months ago

I 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.

• GreenMeansGO 7 months ago

Oh, 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 Agarwal 7 months ago

This 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 Chopra 7 months ago

This was amazing!
Like next level clever...thanks mate!! :D

• Shubham Saraf 8 months ago

What do you mean by exclusive?

• Lumina Mathavan 8 months ago

Good mental exercise. Thank you sir

• Jeffrey Cloete 8 months ago

That was beautiful. .thanks!

• human :3 8 months ago

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

• Yerram Varun 8 months ago

That was awesome! Love your channel

• Rajendra Misir 8 months ago

Terrific 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 rangare 8 months ago

Nice Problem.

• Paul Murray 8 months ago

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

• dunkelheit 8 months ago

Why 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.

• dunkelheit 7 months ago

@π Super J π Thank you so much!

• π Super J π 8 months ago

If 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 Poornash 8 months ago +1

very nice solution

• el tapa 8 months ago

Ahhhh very nice. I didn't knew that tangent identity

• Did you get my solution in email?

• Shanmuga Sundaram 8 months ago

Very nice illustration.Thank you.

• adandap 8 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 Lokii 8 months ago +3

What is your reasoning that alpha and beta are complimentary?

• Sea cucumber 8 months ago

@Slight Lokii Happy to help :)

• Slight Lokii 8 months ago +1

Sea cucumber oh I understand now! Thanks :)

• Sea cucumber 8 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 Les 8 months ago +3

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

• Antonio Banda 8 months ago +5

• Cristhian Martínez 8 months ago +1

Amazing

• Ben Burdick 8 months ago +7

I feel really dumb for not seeing the trig sub when I attempted this problem.

• Santhosh Kumar 8 months ago +7

Thank you very much for the solution. I am big fan of your channel.

• Santhosh Kumar 8 months ago

Sir,
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 Berg 8 months ago

In what cases again i can use trig substitution?
What if the question doesnt tell that 0

• Jerry Tan 8 months ago

Just 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 Berg 8 months ago +1

@Prathmesh Joshi thx. That might be a clue

• Prathmesh Joshi 8 months ago +1

note that -1

• Pete Berg 8 months ago

Well, this open my mind. I hope i can apply this strategy as my tool

• Ducksfan101 8 months ago +54

Okay that’s epic.

• bridogg154 8 months ago

Solve this one usclip.net/video/m0PqvrZbcTo/video.html