A geometry problem by Ishita Bhatia
Geometry
Level
pending
Cos (α - β) = 1 and
cos (α + β) = 1/e, where α, β∈[-π, π].
Find the Pairs of α, β which satisfy both the equations .
(This problem is from JEE-2005)
The answer is 4.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Cos(α-β)=1. This shows that (α-β)=0 or 2π. Now 2π can only be achieved if α=π and β=-π, but this doesn't satisfies the 2nd equation therefore, (α-β)=0 so (α=β). Putting this in second equation we get cos(2α)=1/e. Now we know that -π≤α≤π. So -2π≤2α≤2π. Also cos(2α) is positive therefore, α lies in 1st or 4th quadrant. So 4 values of α exist in the interval [-2π,2π]. As (α=β) so 4 values of β also exist in the interval [-2π,2π]. Hence, in total 4 pairs of β & α exist collectively.