Solve trig equation

**ling_c_0202** · April 18th 2006, 08:16 AM

I can't get the correct ans for this question~~

Solve the equation for 0 deg <= theta <= 360 deg.

$ \sin 5 \theta/ \sin \theta - \ cos 5 \theta/\cos \theta = 2 $

**ThePerfectHacker** · April 18th 2006, 09:22 AM

Originally Posted by ling_c_0202

I can't get the correct ans for this question~~

Solve the equation for 0 deg <= theta <= 360 deg.

$ \sin 5 \theta/ \sin \theta - \ cos 5 \theta/\cos \theta = 2 $

Add fractions (remember commom denominator

):
$\frac{\sin 5x \cos x-\sin x\cos 5x}{\sin x\cos x}=2$
But the numerator is expansion of $\sin (5x-x)=\sin 4x$
Thus, you have,
$\frac{\sin 4x}{\sin x\cos x}=2$
Multiply both sides by a half to get,
$\frac{\sin 4x}{2 \sin x\cos x}=1$
But, $2\sin x\cos x=\sin 2x$ and $\sin 4x=2\sin 2x\cos 2x$ (double angles),
$\frac{2\sin 2x\cos 2x}{\sin 2x}=1$
Notice the $\sin 2x$ kills the other $\sin 2x$ thus,
$2\cos 2x=1$
Thus,
$\cos 2x=1/2$
Thus, all the solutions take the form,
$2x=\left\{ \begin{array} {c}\frac{\pi}{3}+2\pi k\\ -\frac{\pi}{3}+2\pi k$
Thus,
$x=\left\{ \begin{array}{c} \frac{\pi}{6}+\pi k\\ -\frac{\pi}{6}+\pi k$
You need them to be such as, $0\leq x\leq 2\pi$
For the example the top solution is true only for $k=1$ otherwise it gets too big or too small. Substituting $k=1$ for the top solution we get,
$\frac{\pi}{6}+\pi=\frac{7\pi}{6}$
The bottom solution is true only for $k=1,2$ otherwise it gets too big or too small. Substituting $k=1,2$ for the bottom solution we get,
$-\frac{\pi}{6}+\pi=\frac{5\pi}{6}$
$-\frac{\pi}{6}+2\pi=\frac{11\pi}{6}$
Thus,
$x=\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}$

**ling_c_0202** · April 19th 2006, 08:23 AM

but for the step
$ \frac{2\sin 2x\cos 2x}{\sin 2x}=1 $

it is not given in the question sin 2x <> 0,

how come they could kill each other?

is it possible if i 'lost' any solution?

**ThePerfectHacker** · April 19th 2006, 10:41 AM

Originally Posted by ling_c_0202

but for the step
$ \frac{2\sin 2x\cos 2x}{\sin 2x}=1 $

it is not given in the question sin 2x <> 0,

how come they could kill each other?

is it possible if i 'lost' any solution?

Because $\frac{\sin 2x}{\sin 2x}=1$ for defined values of $x$

**ling_c_0202** · April 20th 2006, 06:28 AM

THANK YOU!!

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