The problem involves the composition of functions, specifically determining the value of \( (g \circ f)(x) \) where \( f(x) = \sin^2 x + \sin^2(x + \pi/3) + \cos x \cos(x + \pi/3) \) and \( g(5/4) = 1 \). The challenge arises from the lack of a defined function \( g(x) \) beyond the given point.
Participants are actively engaging with the problem, attempting to simplify \( f(x) \) and questioning the role of the given \( g(5/4) \). There is recognition of the need for careful simplification, and some participants express gratitude for suggestions that guide their thought process.
There is an emphasis on the missing definition of \( g(x) \), which is central to resolving the problem. The discussion reflects uncertainty about how to proceed without additional information about \( g(x) \).
Use trig sum formulas to simplify ## f(x) = sin^2x + sin^2(x+ π/3) + cosxcos(x+ π/3) ##. If you're careful, you can get this simplfied a great deal.Raghav Gupta said:Homework Statement
If ## f(x) = sin^2x + sin^2(x+ π/3) + cosxcos(x+ π/3) ## and $$g(5/4)=1$$, then ##(gof)(x)## is equal to?
Options are
0
1
sinx
None of these
Homework Equations
I know gof(x) is g(f(x)) but here g(x) is not given.
The Attempt at a Solution
Basically I think we have to put f(x) value in g(x) but here g(x) is not given,
What's the use of g(5/4) ?
By simplifying I got,Mark44 said:Use trig sum formulas to simplify ## f(x) = sin^2x + sin^2(x+ π/3) + cosxcos(x+ π/3) ##. If you're careful, you can get this simplfied a great deal.
Yes, that's what I got as well.Raghav Gupta said:Sorry, carefulness was required as you said.
Got on simplifying now 5/4, and then the answer is 1.
Thanks for the motivation of simplifying.