Solve the equation:
My attempt:
let ......1
and .......2
from these:
substitute 1 and 2 into the equation:
substitute and use the identity:
and where
my problem is the , I tried using pythagoras' theorem to find the the length of the adjacent side the then use that to determine the cosine, but I can't work it out.
thanks for any help.
Hello arzeThe problem of extra 'solutions' is always likely to occur when we square things as part of solving an equation, because squaring a real number always 'hides' a possible minus sign by giving a positive value, whatever the sign of the original number.
It's pretty obvious when you do something like this, for instance:
or
All of which is correct, but that doesn't mean that you can reverse the implication arrows and say that both values of are valid solutions to the original equation.
is OK. (This gives .)
But isn't. This gives , and .
So, whereas is a valid solution to the equation
it doesn't necessarily mean that it's a solution to the original equation. gives and , whereas you'll find you need to satisfy the original equation.
The problem arises when we use , because this has the same value, , whether or .
So, we all need to take care with signs when we take squares and square roots!
Grandad