ISL 91
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 $a$.
(angles are in radians)Determine,with proof,the value of $a$.
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Re: ISL 91
My steps:
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Re: ISL 91
Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....
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Re: ISL 91
Who said $a$ is an integer?It's rational only.sakibtanvir wrote:Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....
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- Phlembac Adib Hasan
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Re: ISL 91
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.)SANZEED wrote: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 $a$.
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Re: ISL 91
Phlembac Adib Hasan wrote:
Edited.
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Re: ISL 91
My solution(in brief):
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Re: ISL 91
What's $a_n$ and $b_n$?SANZEED wrote:My solution(in brief):
Re: ISL 91
They are odd integers. Sorry for eliminating.
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Re: ISL 91
\[-1<cos\angle x<1\] for any value of $x$ .Phlembac Adib Hasan wrote:Who said $a$ is an integer?It's rational only.sakibtanvir wrote:Converting radian into degree,we get,\[cos(540a)+2cos(360a)=0\]
\[Or,cos(180a)+2cos0=0\]
\[Or,cos(180a)=-2\]
Contradiction.....
So there is no real value of $a$.
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.