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What is the period of $$\frac{7\sin x + 5\cos x}{7\sin{2x} + 11\cos x}$$

What should I do here? I don't even know where to start from. Please help me by giving me a hint!!

Thanks.

amWhy
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Gummy bears
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  • The period of the entire function is going to be the largest period of all of the trigonometric functions. – Brad Aug 01 '14 at 19:43
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    Well, it's really going to be the least common multiple of all the periods of the trigonometric functions, barring cancellation. – davidlowryduda Aug 01 '14 at 19:46
  • @mixedmath Two different answers? Help? – Gummy bears Aug 01 '14 at 19:47
  • In this particular case, both answers coincide. But Brad's statement isn't true in general. It was an easy, honest mistake. – davidlowryduda Aug 01 '14 at 19:48
  • Sooo what do you mean by the least common multiple @mixedmath – Gummy bears Aug 01 '14 at 19:49
  • If you have some numbers $a_1,a_2,\ldots,a_n$. The least common multiple is the smallest number so that each $a_i$ divides it. For instance, the least common multiple of $2$, $3$, and $8$ is $24$ because it is the smallest number that all of $2$, $3$, and $8$ divide. – J126 Aug 01 '14 at 20:12
  • Try replacing $x$ with $x + 2\pi$ and see if you can show that the expression simplifies to original one. Show this isn't true if $x$ is replaced with $x + \pi$ . ( When in doubt, graph the expression to see what is going on -- yes, I know you can't do this in a test situation.) – Alan Aug 01 '14 at 20:25
  • @JoeJohnson126 Well that was pretty obvious. But don't understand it in terms of trigonometric functions. – Gummy bears Aug 01 '14 at 20:26
  • @Alan How exactly will that help? I need to find the period of the function? – Gummy bears Aug 01 '14 at 20:26
  • ^If you can show that $f(x+2\pi) = f(x)$, then you have shown that the function $f$ is $2\pi$-periodic. – JimmyK4542 Aug 01 '14 at 20:28
  • Honestly , none of the tips offered seemed evident to me, so I graphed the function. I'm trying to make things as simple as possible. – Alan Aug 01 '14 at 20:31
  • @JimmyK4542 I see. Let me try. But there has to be a more definite way? Rather than just trial and error? – Gummy bears Aug 01 '14 at 20:31
  • @Alan Would $\pi$ be the correct answer? – Gummy bears Aug 01 '14 at 20:32
  • @Gummybears Nope. – Hakim Aug 01 '14 at 20:33
  • Hmmmmm....... I do get the same function over again though? @Hakim – Gummy bears Aug 01 '14 at 20:34
  • If you find that the expression can be simplified to the original when $x$ is replaced by $x + \pi$ , yes. – Alan Aug 01 '14 at 20:34
  • Arghhh. This is just trial and error. Anyone with a more definite answer? – Gummy bears Aug 01 '14 at 20:36
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    @Gummybears You made a mistake when calculating $f(x+2\pi)$, the correct answer is $2\pi$. I suspect that you're mistake was with the $7\sin2x$ term in the numerator, using the substitution $x\leadsto x+\pi$ yields $7\sin(2x+2\pi)$ and not $7\sin(2x+\pi)$. – Hakim Aug 01 '14 at 20:39

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