2001 Paper 2 Q4

Year: 2001
Paper: 2
Question Number: 4

Course: LFM Pure
Section: Trigonometry 2

Difficulty: 1600.0 Banger: 1484.0

trigonometry sin nx quadratic in cos roots of equations discriminant factorization proof trigonometric identities

Problem

Let $$ \f(x) = P \, {\sin x} + Q\, {\sin 2x} + R\, {\sin 3x} \;. $$ Show that if $Q^2 < 4R(P-R)$, then the only values of $x$ for which $\f(x) = 0$ are given by $x=m\pi$, where $m$ is an integer. \newline [You may assume that $\sin 3x = \sin x(4\cos^2 x -1)$.] Now let $$ \g(x) = {\sin 2nx} + {\sin 4nx} - {\sin 6nx}, $$ where $n$ is a positive integer and $0 < x < \frac{1}{2}\pi $. Find an expression for the largest root of the equation $\g(x)=0$, distinguishing between the cases where $n$ is even and $n$ is odd.

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Difficulty Rating: 1600.0

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Banger Rating: 1484.0

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Problem source

Let
$$
\f(x) = P \, {\sin x} + Q\, {\sin 2x} + R\, {\sin 3x} \;. 
$$
Show that if $Q^2 < 4R(P-R)$, 
then the only values of $x$ for which $\f(x) = 0$ are given by $x=m\pi$, where $m$ is
an integer.
\newline
[You may assume that $\sin 3x  =  \sin x(4\cos^2 x -1)$.]
Now let 
$$
\g(x) =  {\sin 2nx} + {\sin 4nx} - {\sin 6nx},
$$
where $n$ is a positive integer and $0 < x < \frac{1}{2}\pi $.
Find an expression for the largest root of the equation
$\g(x)=0$, distinguishing between the
cases where $n$ is even and $n$ is odd.

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