Makes complete sense thank you so much!!!
My only question is how come we respect domains only sometimes...
For example
now sin©¬/cos©¬ = (1/a)/(¡î3/a) = 1/¡î3
tan©¬ = 1/¡î3
©¬ = 30¨¬ or pi/6
then sin 30¨¬ = 1/a
1/2 = 1/a ---> a = 2
so our original ¡î3 sin x + cos x = 1 becomes
2sin(x+30¨¬) = 1
sin(x+30¨¬) = 1/2
x+30 = 0 or x+30 = 150¨¬
x = 0¨¬ or 120¨¬
the period of sin(x+30) is 360¨¬ so we can add 360 to any answer as long as that keeps us in our domain
so x = 0, 120 , or 360
just like I had before
Makes sense... perfect sense but...
notice when we decided on beta
tan©¬ = 1/¡î3
©¬ = 30¨¬ or pi/6
we used pi/6 not (7 pi)/6 because of the restriction of tangent is between (3 pi)/2 and pi/2
(7 pi)/6 is a perfect solution for beta but it fall out of the restrictios for tangent so we ignore it...
but what does not make sense
sin(x+30¨¬) = 1/2
x+30 = 0 or x+30 = 150¨¬
x = 0¨¬ or 120¨¬
the period of sin(x+30) is 360¨¬ so we can add 360 to any answer as long as that keeps us in our domain
so x = 0, 120 , or 360
makes sense makes sense...
but we did not respect the domain here and i don't know why but we still get the right answer...
how come???
see
sin(x+30¨¬) = 1/2
sin^-1 1/2
has two solutions pi/6 or(5 pi)/6 and the domain of sine is between (3 pi)/2 and pi/2 and as we can see the solution (5 pi)/6 falls out of this restriction...
x+30 = 30 or x+30 = 150¨¬
which gives us the right answers...
of 0 or (2 pi)/3
even though (5 pi)/6 was out of the domian of sin and we did not respect the domian here
but we respected the domain for tangent...
I just have to know why this is so...
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