Trigonometry problem

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Melanie Leconte
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#1
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#1
Hi, I would really appreciate your help with this problem :

arctan x + arccos x = pi/4


Thank you
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Mr M
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#2
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(Original post by Melanie Leconte)
Hi, I would really appreciate your help with this problem :

arctan x + arccos x = pi/4


Thank you
What have you tried?
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math42
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#3
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You can use compound angle formulae; for instance try taking the tangent of both sides of the equation.
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Melanie Leconte
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#4
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Thank you. I have tried using some of the trigonometric identities including the sum & difference formulas for sin, cos and tan.
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atsruser
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#5
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#5
(Original post by Melanie Leconte)
Hi, I would really appreciate your help with this problem :

arctan x + arccos x = pi/4


Thank you
Assuming that you want to solve for x, then this question would be easier if it were all in terms of, say, arctan. So suppose that \phi=cos^{-1} x and try to find an expression that introduces \tan somehow. (Hint: draw an appropriate right angled triangle).
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Melanie Leconte
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#6
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#6
(Original post by Mr M)
What have you tried?

Thank you. I have tried using some of the trigonometric identities including the sum & difference formulas for sin, cos and tan.
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Mr M
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#7
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(Original post by Melanie Leconte)
Thank you. I have tried using some of the trigonometric identities including the sum & difference formulas for sin, cos and tan.
The easiest way is to tan both sides and use the expansion of tan (A+B).

Look at the identities on this page:

http://en.wikipedia.org/wiki/Inverse...tric_functions
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atsruser
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#8
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(Original post by Melanie Leconte)
Hi, I would really appreciate your help with this problem :

arctan x + arccos x = pi/4


Thank you
Did you manage to solve this? I get the only solution to be x=1 but that's by way of a messy quartic, one of whose roots I had to find via Wolfram alpha, and which can be ruled out by a non-trivial argument about the behaviour of \arctan x +\arccos x over [-1,1] (or which in fact I did by checking its graph)

This seems far too tricky for A level - is it a STEP question?
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Melanie Leconte
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#9
Report Thread starter 7 years ago
#9
(Original post by atsruser)
Did you manage to solve this? I get the only solution to be x=1 but that's by way of a messy quartic, one of whose roots I had to find via Wolfram alpha, and which can be ruled out by a non-trivial argument about the behaviour of \arctan x +\arccos x over [-1,1] (or which in fact I did by checking its graph)

This seems far too tricky for A level - is it a STEP question?

I don't know if it is from step paper. I was set it as a brain teaser for year 11.
I have managed to solve it and I can see what you mean about the quartic
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atsruser
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#10
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#10
(Original post by Melanie Leconte)
I don't know if it is from step paper. I was set it as a brain teaser for year 11.
I have managed to solve it and I can see what you mean about the quartic
Where do you study that you do this in year 11? Hogwarts?
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