Question

Simplify the following by using the geometry of an appropriate triangle

tan(sin-1(x))
and
cos(tan-1(x))

Answer 1

there is a picture of a triangle and an angle \(\theta\) with \(\sin(\theta)=x\) thinking of sine as opposite over hypotenuse
what is missing is the third side which you find via pythagoras

Answer 2

Ok, so sin(theta) = opp./hyp. Then how do we use Pythag to find the '?' side if we dont have x? Or we use x as it is to plug in?

Answer 3

there is an x in your answer

Answer 4

\[x^2+?^2=1\] is a start

Answer 5

clear or no?

Answer 6

Clearer yes. So the third side will equal sqrt(1-x^2), right?

Answer 7

yes

Answer 8

with a tiny bit of practice this will be immediate

Answer 9

now you can take any other trig ratio you like, including "opposite over adjacent"

Answer 10

So now we need to find the inverse of sin, right? so sin-1=hyp/opp right

Answer 11

hold the phone
you were trying to find this composition \[\tan(\sin^{-1}(x))\] right? in terms of x alone

Answer 12

Yes. to do that do we find inverse sin, and plug that term into tan(x)?

Answer 13

no

Answer 14

the angle IS the arcsine of x , you want the tangent of that angle
do "opposite over adjacent" for the triangle labelled above

Answer 15

or rather "its"

Answer 16

Oh ok thank you. So we end up with x/(sqrt(1-x^2)). Is that the final simplified version?

Answer 17

yes

Answer 18

Now for the second part, when finding cos(tan-1(x)), are we able to use the same original triangle setup or do the positions of the 1 and the x need to change for some reason?

Answer 19

you have to change the triangle so the x is the "opposite" and the 1 is the adjacent, since tangent is that

Answer 20

then solve for the hypotenuse
it is similar, but different

Answer 21

Ok. And how did you know that the x needs to be the opposite, and the 1 must be adjacent, rather than that being switched or something maybe?

Answer 22

I thought it might be if we're finding the inverse of tanx, the "opposite over adjacent" must give us 1/x. But no?

Answer 23

no

Answer 24

inverse does not mean reciprocal

Answer 25

Ok - because an arctangent is simply the angle of the tangent sides. Makes sense.

Answer 26

So the cos(tan-1(x)) is 1/1+x^2?

Answer 27

Because i used Pythag to solve for the third side and I found the adj./hyp.

Answer 28

you should have a square root there