Trigonometric equation

Squish

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Feb 16, 2019
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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.
 
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.
 
it'll be:
2sin(x) - 2sin(x)cos(x) = 4/pi

But I can't solve that either.
 
The pi in the denominator seems odd. Please check that you copied correctly; does it really say [MATH]10\sin(x)-5\sin(2x)=\frac{20}{\pi}[/MATH]?
 
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.
 
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?
 
Where did this problem come from? Are you allowed to use a graphing calculator? How about estimates?
 
… 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)

?
 
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. :)
 
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.
 
I get:

[MATH]x\approx1.23822452160548715648[/MATH]
[MATH]x\approx2.8081205503453334222[/MATH]
 
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".
 
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