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Math Help - Cosine Proof

  1. #1
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    Cosine Proof

    I have to answer the following and support it with a full proof.

    Is cos(5) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).
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  2. #2
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    Quote Originally Posted by harold View Post
    I have to answer the following and support it with a full proof.

    Is cos(5) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).
    \cos (3x)=\cos (x+2x)
    \cos x\cos 2x-\sin x\sin 2x
    \cos x(\cos ^2 x-\sin^2 x)-\sin x (2\sin x\cos x)
    \cos^3 x-3\cos x \sin^2 x
    And,
    \sin^2 x=1-\cos ^2x
    So if a=\cos x is rational
    Then,
    a^3-3 a(1-a^2) is also rational.
    ---
    If \cos (5^o) is rational.
    Then,
    \cos (15^o) is rational.
    Then,
    \cos (45^o)=\frac{\sqrt{2}}{2} is rational.
    But that is not true.
    Contradiction.
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  3. #3
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    Hi THP,

    I was thinking the exact same thing by showing cos(45) but it still doesn't prove it since we need to show if cos x is rational, then cos 3x is rational....then show cos 5 is irrat. based on the previous two proofs...?
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  4. #4
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    Quote Originally Posted by harold View Post
    Hi THP,

    I was thinking the exact same thing by showing cos(45) but it still doesn't prove it since we need to show if cos x is rational, then cos 3x is rational....then show cos 5 is irrat. based on the previous two proofs...?
    Yes it does.

    If assume \cos 5^o
    Is rational and you arrived at a contradiction.

    If a=\cos x is a rational number then,
    \cos 3x=a^3-3a(1-a^2) is a rational number.
    Because when you add or multiply rationals you get a rational.
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