What have you tried? Please show us your work. Did you try 1* 1/2* 1/4? How about (1/2)^3... If you want to guess the answer then these are the things you need to try. If this fails then it is time for heavy algebra.I am trying to find an efficient way of solving
cosxcos4xcos8x=1/8, 0 to 2pi, i am thinking about the product of three trig expressions and what must be the case to equal 1/8.
The alternatives feel too heavy algebraically?
What have you tried? Please show us your work. Did you try 1* 1/2* 1/4? How about (1/2)^3... If you want to guess the answer then these are the things you need to try. If this fails then it is time for heavy algebra.
Yes, i was trying different possible products! and wondering if there is some kind of symmetry to exploit.
I have tried sketching it for inspiration.
After learning the trick please report back here!Yes, a friend who teaches maths sent me this and asked if i can spot a nice trick?
But after musing on it for a while, like you, i haven't made much progress.
I will get back to him in a bit for the 'trick' but wanted to thrash it around on here first!
Just thinking out loud. How about letting u=4x, 2x=u/2 and 8x = 2u. Will it simplify using the half angle and double angle formulas? I suspect not, since Dr P tried using various identities to simplify the equation with no luck.I am trying to find an efficient way of solving
cosxcos4xcos8x=1/8, 0 to 2pi, i am thinking about the product of three trig expressions and what must be the case to equal 1/8.
The alternatives feel too heavy algebraically?
I have been given a hint:
Try multiplying by sinx!
I can see how that would help (up to a point) for a slightly different equation; but I don't yet see how to use it here.
Just to make sure, did you copy the equation correctly? We have it as
cos(x) cos(4x) cos(8x) = 1/8
Yes!
My mistake. So sorry!
What a mess up.
The original question is cosx cos2x cos4x = 1/8
You are an amazing mathematician.To be more precise, the trick works very nicely for cos(x) cos(2x) cos(4x) = 1/8. I might even have thought of it in that case. (But probably not.)
I'll second that!You are an amazing mathematician.
Now then follow the hint and show us what do you get.
so multiply by sinx
sinx cosx cos2x cos4x = 1/8 sinx
1/2 sin2x cos2x cos4x = 1/8 sinx
(1/2)(1/2)sin4x cos4x = 1/8 sinx
(1/8) sin8x = (1/8) sinx
so sin8x = sinx
i think we can do some general solution stuff from here?
Why not just check these values in the original equation to see if they work? Can you show us your work that lead to 8x = x+ 360n or 180-x +360nso would this lead to 8x = x+ 360n or 180-x +360n
x = (360/7)n or x = 20 +40n
Is that correct?
so would this lead to 8x = x+ 360n or 180-x +360n
x = (360/7)n or x = 20 +40n
Is that correct?