A Nice Trigonometric Problem | Math Olympiad
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- čas přidán 13. 09. 2024
- A Nice Trigonometric Problem | Math Olympiad
Welcome to another exciting Trigonometric Problem! In this video, we explore a nice trigonometric simplification problem from the Math Olympiad. This problem will test your understanding of trigonometric identities and your ability to simplify trig expressions. Can you simplify this trig expression? Watch the video, try it yourself, and share your solution in the comments below. Don’t forget to like, share, and subscribe for more Math Olympiad challenges and math problem-solving videos!
Topics Covered:
Trigonometry
Expressions
Trigonometric identities
Math Olympiad
Simplification
Math Olympiad Preparation
Math Tutorial
Find unknown
Rationalization
Trigonometric expression
How to Evaluate Trig Expression
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Thank you for watching!!
This is both a nice problem and a nice presentation.
Note ∏/6+∏/24 = ∏/3-∏/8 and tan ∏/3 = √3 and tan ∏/8 = √2-1 (using tan∏/4=1). So, tan [ ∏/6+∏/24] = tan[∏/3-∏/8] = [√3-( √2-1)]/[1+√6-√3]. But tan( ∏/6+∏/24) simplifies to [4-(√3-1)a]/[(√3+1)a]. So, [4-(√3-1)a]/[(√3+1)a] = [√3-( √2-1)]/[1+√6-√3] which yields, after some algebra, a= √2.
a= √2
Very complicated solution.
There's a much easier one expressing a in terms of cot(Pi/24) and then cot(Pi/24)=cos(Pi/24) / sin(Pi/24).
Then the numerator is equal to sqr(3) * cos(Pi/24) - sin(Pi/24), which is 2 * sin(7*Pi/24)
And the denominator is cos(Pi/24) + sin(Pi/24) which is sqr(2) * sin(7*Pi/24)
Hense a=sqr(2)
A=√2
let a=pi/24, c=cos(a),s=sin(a), r3 = sqrt(3). Problem is rewritten (1+x)/(r3-x) = c/s => x= (r3*c-s)/(s+c) = r3 - (r3+1)s/(s+c)
or x = r3 - (r3+1)/(1+cot(a)) ~1.4142. (OK, maybe not as nice as your method)
cotPi/24 cot Pi/4^6 cotPi/2^23^2 cotPi/1^23^1 23 (cotPi ➖ 3cotPi+2). { 2a+2a ➖}=2a^2/(3)^2 ➖ (a)^2= 2a^2/{9 ➖ a^2}={ 2a^2/7}= 3 .1a^2 3^1.1^1a^2^1 3a^2 (a ➖ 3a+2) .
Με ολο το σεβασμο θελω να σας παρακαλεσω κατι. Να γραφετε το 3,14...σαν π και οχι σαν ^και απο πανω μια παυλα _. ελληνικο π. Και παλι μπραβο για τις ασκησεις σας. Ευχαριστω.
All this algebra is not needed cot(pi/24) is a constant, just solve for a in terms of cot(pi/24) simplification is not always needed.
Yes, that will give the solution for x if cot(pi/24) is given.
@@ericerpelding2348 cot(pi/24) is not a variable it is given open your eyes
Made a mistake at 3.30