Trigonometric equation

Squish

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I'm stuck in solving for x on this one:

10sin(x)-5sin(2x)=20/pi , where 0<=x<=pi

I've got a midterm tomorrow and the answer for this one is not in the back of my book. Please help.
 

tkhunny

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Have you considered the sine's double angle formula? You should memorize this important formula.

Also, you may wish to divide by 5, just to make your life a little easier. Up to you. Sometimes, it's convenient not to simplify as you go along.
 

Squish

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it'll be:
2sin(x) - 2sin(x)cos(x) = 4/pi

But I can't solve that either.
 

Dr.Peterson

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The pi in the denominator seems odd. Please check that you copied correctly; does it really say \(\displaystyle 10\sin(x)-5\sin(2x)=\frac{20}{\pi}\)?
 

Squish

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The pi in the denominator seems odd. Please check that you copied correctly; does it really say \(\displaystyle 10\sin(x)-5\sin(2x)=\frac{20}{\pi}\)?
Yes, I am positively sure that is how it is written.
 

Jomo

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Squish

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Now you can divide by 2.
Alright, so the equation would then be
sin(x) - sin(x)cos(x)=2/pi
but still there's not much to do here; as taking sin(x) as a common factor and then trying to divide it out of the equation won't work as the 2/pi would still hold it.
 

Dr.Peterson

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Well, we could replace cos(x) with sqrt(1 - sin^2(x)) and solve the resulting radical equation in sin(x); but that turns into a quartic equation that, with the irrational numbers involved, doesn't seem very workable.

So here's a new question: In your context, is it possible that you are expected to find approximate solutions using technology, rather than exact solutions by algebraic means? What else would I learn if I grabbed your book from you and looked through the context? Are any special methods needed for other problems nearby?
 

Jomo

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Where did this problem come from? Are you allowed to use a graphing calculator? How about estimates?
 

Otis

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Otis

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… doesn't seem very workable …is it possible that you are expected to find approximate solutions using technology, rather than exact solutions …
That was my thought, also. There are two solutions, and (according to MVR5) the exact form of the larger solution looks like this:

arctan(1/6sqrt(6)sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))-1/6sqrt((-6sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)-72sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))+36sqrt(6)(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3))/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3))))),-1+1/2Pi(1/6sqrt(6)sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))-1/6sqrt((-6sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)-72sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)))+36sqrt(6)(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3))/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3)sqrt(((27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(2/3)+12)/(Pi(27Pi+3sqrt(3)sqrt(-64+27Pi^2))^(1/3))))))^3)

😎
 

MarkFL

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When I initially looked at this question, after some consideration and then looking at the HUGE solutions given by W|A, I concluded that I would use a numeric root finding technique (like the Newton-Raphson method), to approximate the two roots on the given domain. I would use a graph to generate the initial seed values. :)
 

Squish

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Sorry for the confusion, I think I should've given a lot more context. So this part of a question on my 'webassign' assignment for calculus 2. The original question was asking me to get the average f (x) of the equation which is 20/pi by obtaining the product of (1/(b-a))(the definite integral of the above equation from b to a) where a is 0 and b is pi. The definite integration is equal to 20, so dividing it by pi would make it 20/pi. The part which I am asking about DOES expect a solution of two irrational numbers approximated to 5 decimal points. That should explain why it doesn't look like a typical highschool trigonometry question. I think I am allowed to use a graphing calculator on this assignment, but I don't own one at the moment; I am not allowed to use one in my calculus 2 exams.
 

MarkFL

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I get:

\(\displaystyle x\approx1.23822452160548715648\)

\(\displaystyle x\approx2.8081205503453334222\)
 

Dr.Peterson

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There are many online equivalents to a graphing calculator, which you can use for homework; a couple have been mentioned. A simple one is Desmos.com, on which you can graph y = 10sin(x)-5sin(2x) and y = 20/pi, then click on each intersection and see the coordinates (to only three decimal places). For more detailed and complicated answers, go to WolframAlpha.com and enter "solve 10sin(x)-5sin(2x) = 20/pi".
 

Squish

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Thank you so much
 
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