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3 sin(theta) = tan(theta)

Hi all,

I have a question asking to solve this equation (rounded to 2 d.p.) in the domain -pi <(or equal to) theta <(or equal to) pi.
I do know how to use the identity tan(theta) = sin(theta)/cos(theta), which leads to cos(theta) = 1/3, so theta = -1.23 or 1.23.
However, my book suggests three other answers as well, which are -pi, 0 and pi. Could anyone explain to me why those answers as well? In any equation of this type, what do I have to remember to cover all possibilities?

Many thanks
Reply 1
I guess at some point in your working you divided by sin(theta) because you had it on both sides. But sin(theta) could be zero.

Any time you divide both sides of the equation by a function f(x) of the variable x, ask yourself "could I be dividing by zero here?"

If so, branch your analysis into two cases. Case 1: when it is zero, you can search separately for solutions to f(x) = 0, and case 2: when it is not zero, and you can divide by it and continue as before.


As a side-note, I don't agree that +/- pi are solutions, since tan(theta) is not defined at these points.
(edited 8 years ago)
Reply 2
Thank you for your answer Mik.

You're right, I divided by sin(theta). Let's call 'theta' 'x' to simplify writing it down.

3sinx = tanx
3sinx = sinx/cosx
3cosx = sinx/sinx
3cosx = 1
cosx = 1/3
cos^-1(1/3) = -1.23 and 1.23

tanx is actually defined for x = pi. It isn't for x = 1/2pi and every interval of pi from that point. Both the sin function and the tan function equal to 0 when x = -pi, 0 and pi. Would that be the reason? My book seems to have skipped this bit in the explanations.
Since you're solving for
3sin(x) = tan(x) = sin(x)/cos(x)
When sin(x) = 0 the equation is still valid and so those solutions for sin(x) = 0 needs to be covered.

I guess there isn't a checklist to look out for when doing these, but I would say just watch out for zeroes of functions (especially when multiplying and dividing) since they're usually the ones giving extra solutions.

Another thing is to check the domain and remember the +n*pi when appropriate.

(I need a bit of help at my own problem too, would be wonderful if you could take a look at my thread :wink: )
(edited 8 years ago)
Reply 4
Original post by oniisanitstoobig
Since you're solving for
3sin(x) = tan(x) = sin(x)/cos(x)
When sin(x) = 0 the equation is still valid and so those solutions for sin(x) = 0 needs to be covered.

I guess there isn't a checklist to look out for when doing these, but I would say just watch out for zeroes of functions (especially when multiplying and dividing) since they're usually the ones giving extra solutions.

Another thing is to check the domain and remember the +n*pi when appropriate.

(I need a bit of help at my own problem too, would be wonderful if you could take a look at my thread :wink: )


Thanks. It does make sense indeed. Just some extra little things we need to keep in mind.
How do I find your thread? I don't know my way around this website yet.
If you rollover my username on the left you'll see a link to my recent posts, there you'll find something titled along the lines of 'choosing solutions for trig functions', or, if you click on my profile you willl get to the same link under the tab - my stats.

I joined the site yesterday just to ask that question lol, trying to find my way round just like you.
Reply 6
Original post by oniisanitstoobig
If you rollover my username on the left you'll see a link to my recent posts, there you'll find something titled along the lines of 'choosing solutions for trig functions', or, if you click on my profile you willl get to the same link under the tab - my stats.

I joined the site yesterday just to ask that question lol, trying to find my way round just like you.


Sorry mate, I've been away for a while. I'll have a look today.

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