Question

\[ \cos(x) + \cos(y) - \cos (x+y) = \frac{3}{2}\]

Answer 1

Then what?

Answer 2

solve for y, sorry

Answer 3

gonna read this later, bbl 2 hours *-*

Answer 4

<--will reply in 2 hours

Answer 5

i think not enough information for solve it

Answer 6

@nphuongsun93 What you tried? (just asking)

Answer 7

\(\cos x + \cos y - \cos x \cos y + \sin x \sin y=\frac{3}{2}\)

Answer 8

@asnaseer sir any ideas?

Answer 9

@hartnn

Answer 10

i thought of maybe using cos(a) = [e^(ia) + e^(-ia)] / 2
but i dont think it works either

Answer 11

Maybe try this:\[\cos(x)+\cos(y)=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})\]and:\[\cos(x+y)=2\cos^2(\frac{x+y}{2})-1\]That will lead to:\[\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})-\cos^2(\frac{x+y}{2})=\frac{1}{4}\]

Answer 12

which is a quadratic in \(\cos(\frac{x+y}{2})\) - although it also contains a \(\cos(\frac{x-y}{2})\)

Answer 13

Solving the quadratic leads to:\[\cos(\frac{x+y}{2})=\frac{\cos(\frac{x-y}{2})\pm\sqrt{\cos^2(\frac{x-y}{2})-1}}{2}\]

Answer 14

Which simplifies to:\[\cos(\frac{x+y}{2})=\frac{\cos(\frac{x-y}{2})\pm\sin(\frac{x-y}{2})}{2}\]

Answer 15

Don't know if this is helpful or not but I have to go now - good luck!

Answer 16

Thanks a lot @asnaseer i get it, that's just what i needed

Answer 17

@nphuongsun93 , what and how did you get the answer?

Answer 18

\[\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})-\cos^2(\frac{x+y}{2})-0.25 = 0\]
then let \[a = \cos(\frac{x+y}{2}) \space and \space b = \cos ( \frac{x-y}{2})\]
\[ab - a^2 - 0.25=0\]
taking this as a quadratic in p, b^2 -1 has to be positive to get real solution for a, but the highest value of b = cos((x-y)/2) is 1 (angles), so the only real solution is b =1 or b^2 -1 = 0
\[b= \cos(\frac{x-y}{2})=1 \space or \space \frac{x-y}{2}=0 \space or \space x=y\]