Question

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...

Answer 1

I like how Jishka adds random symbols to my posts lol awsome :O