how do i decompose a function using trig of an arc trig function. Ex. Square root of 1-x^2

Question

loser66 · Answer

don't understand

anonymous · Answer

my text book says "decompose each of the following algebraic functions by using it as a trig function of an arctrig function

anonymous · Answer

$$\sqrt{1- {x}^{1}}$$

alekos · Answer

standby, i'm working on it

loser66 · Answer

thanks @alekos

anonymous · Answer

thanks! and if you can explain it, that would help so much!

alekos · Answer

this is quite tricky but here it goes...


 y = sqrt(1 - x^2)

we let x = sin(theta)  i.e. theta = arcsinx

so we have 

y = sqrt[1 - sin^2(theta)]

follow so far?

anonymous · Answer

yes

alekos · Answer

OK, from this we can see that y = cos(theta)

due to the relationship   sin^2(theta) + cos^2(theta) = 1

so we have 

cos(theta) = sqrt(1-x^2)

but theta = arcsinx from above

and hence we have cos(arcsinx) = sqrt(1-x^2)

anonymous · Answer

holy crap... i never saw it like that

alekos · Answer

we can also deduce that  sin(arccosx) = sqrt(1-x^2)

alekos · Answer

yeah, that's how its done. quite tricky and hard to get your head around

anonymous · Answer

yea, my book, nor my teacher never helps... thanks!

alekos · Answer

thats ok, your welcome