Trouble in Trigonometry

For discussing Olympiad Level Algebra (and Inequality) problems
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sowmitra
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Trouble in Trigonometry

Unread post by sowmitra » Fri Jan 17, 2014 7:01 pm

$\displaystyle \tan \theta = \sqrt{3}-4\sin 24^\circ$
where, $0<\theta < 90^\circ$.
Find the measure of $\theta$ in degrees.
"Rhythm is mathematics of the sub-conscious."
Some-Angle Related Problems;

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Thanic Nur Samin
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Re: Trouble in Trigonometry

Unread post by Thanic Nur Samin » Tue Jan 21, 2014 11:20 pm

$sin5x$
$= sin5x + sinx - sinx$
$= 2 sin3x cos2x - sinx$
$= 2 (3sinx - 4sin^3 x) (1 - 2sin^2 x) - sinx$
$= 2 (8sin^5 x - 10sin^3 x + 3sinx) - sinx$
$= 16sin^5 x - 20sin^3 x + 5sinx$

Putting x=6, we get:
$sin30= 16sin^5 6 - 20sin^3 6 + 5sin6$
Solving the equation, we get, $sin6=.105...$,so $cos6=\sqrt{1-sin^2 6}=.995...$,$tan 6=\frac{sin6}{cos6}=.105... .....(i)$

Putting x=12, we get:
$sin60= 16sin^5 12 - 20sin^3 12 + 5sin12$
Solving the equation, we get, $sin12=.208...$,so $cos12=\sqrt{1-sin^2 12}=.978...$

So $sin24=2sin12cos12=.407...$

Now,
$tan\theta=\sqrt{3}-4sin24=.105...=tan 6$[from$i$]
So,$\theta=6$
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

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sowmitra
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Re: Trouble in Trigonometry

Unread post by sowmitra » Wed Feb 05, 2014 12:05 am

In this solution, you did some pretty ugly calculations, which are very tough to do without a calculator.
For example, how did you solve the equations for $\sin{6^\circ}$ and $\sin{12^\circ}$ ??
Would you please elaborate...?
"Rhythm is mathematics of the sub-conscious."
Some-Angle Related Problems;

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Thanic Nur Samin
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Joined:Sun Dec 01, 2013 11:02 am

Re: Trouble in Trigonometry

Unread post by Thanic Nur Samin » Sun Feb 09, 2014 1:24 pm

Binary search.
It took me a long time to calculate.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

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