ISL 91

[SANZEED]

Let $a$ be a rational number with $0\leq a\leq 1$. Suppose that \[cos 3\pi a+2 cos 2\pi a=0\].
(angles are in radians)Determine,with proof,the value of $a$.

[SANZEED]

My steps:

Derive an equation.
consider two cases.
One will satisfy the claim.Prove the other to be contradictory.

[sakibtanvir]

Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....

[Phlembac Adib Hasan]

Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....

Who said $a$ is an integer?It's rational only.

[Phlembac Adib Hasan]

Let $a$ be a rational number with $0\leq a\geq 1$. Suppose that \[cos 3\pi a+2 cos 2\pi a=0\].
(angles are in radians)Determine,with proof,the value of $a$.

Edit this.I'm surprised that this problem was posted a long time ago.Why nobody noticed this?(I've visited this topic just now.)

[*Mahi*]

Let $a$ be a rational number with $0\leq a\geq 1$. Suppose that \[cos 3\pi a+2 cos 2\pi a=0\].
(angles are in radians)Determine,with proof,the value of $a$.

Edit this.I'm surprised that this problem was posted a long time ago.Why nobody noticed this?(I've visited this topic just now.)

Edited.

[SANZEED]

My solution(in brief):

Set $x=cos \pi a$. Then the equation is equivalent to $4x^{3}+4x^{2}-3x-2=0$. This turns to $(2x+1)(2x^{2}+x-2)=0$. For the case $2x+1=0\Rightarrow cos \pi a=-\frac{1}{2}\Rightarrow a=\frac{2}{3}$. Now it remains,from further calculation, to prove that for every integer $n\geq 0,cos (2^{n}\pi a)=\frac{a_{n}+b_{n}\sqrt 17}{4}$. This is easily proved by induction.

[Phlembac Adib Hasan]

My solution(in brief):

Set $x=cos \pi a$. Then the equation is equivalent to $4x^{3}+4x^{2}-3x-2=0$. This turns to $(2x+1)(2x^{2}+x-2)=0$. For the case $2x+1=0\Rightarrow cos \pi a=-\frac{1}{2}\Rightarrow a=\frac{2}{3}$. Now it remains,from further calculation, to prove that for every integer $n\geq 0,cos (2^{n}\pi a)=$$\frac{a_{n}+b_{n}\sqrt 17}{4}$. This is easily proved by induction.

What's $a_n$ and $b_n$?

[SANZEED]

They are odd integers. Sorry for eliminating.

[sakibtanvir]

Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....

Who said $a$ is an integer?It's rational only.

\[-1<cos\angle x<1\] for any value of $x$ .
So there is no real value of $a$.