A starting note: In the formulae 15 to 17, 
really mean the inverse trigonometric funcitons. That is, 
. See page 86, 87 in the Openstax textbook.
To check formulae 15, 16, 17, one need to compute derivatives for inverse trigonometric functions.
Topic today: review of inverse funcitons.
A function 
consists of three ingredients: a set 
of inputs (domain), a set 
of outputs (range), and a rule 
for assign each input to exactly one output.
The inverse function goes from the set 
to 
.
Slogan: The outputs of 
turns into the inputs of 
; the inputs of 
turns into the outputs of 
.
Notational convention: Instead of writing 
%3Dx#card=math&code=%5Csin%5E%7B-1%7D%28y%29%3Dx&id=Ra6gL), we still denote 
%20%3D%20%5Carcsin(x)#card=math&code=y%20%3D%20%5Csin%5E%7B-1%7D%28x%29%20%3D%20%5Carcsin%28x%29&id=dLAUi). (using 
as variable and 
as function)
Examples of inverse funcitons.
(1) . 
%3Dx-2#card=math&code=f%28x%29%3Dx-2&id=yl2Gc) . Find 
(x) .
Solution: Denote 
%3Dx-2#card=math&code=y%3Df%28x%29%3Dx-2&id=E0lZM). Then 
.
Thus if 
%3Dy%2B2#card=math&code=f%5E%7B-1%7D%28y%29%3Dy%2B2&id=qJ3NX), then we can check that 
)%3Dy%2B2%3Dx-2%2B2%3Dx#card=math&code=f%5E%7B-1%7D%28f%28x%29%29%3Dy%2B2%3Dx-2%2B2%3Dx&id=RV4YU).
And 
)%3Df(y%2B2)%3D(y%2B2)-2%3Dy#card=math&code=f%28f%5E%7B-1%7D%28y%29%29%3Df%28y%2B2%29%3D%28y%2B2%29-2%3Dy&id=gDYg4).
Conclusion: %3Dy%2B2#card=math&code=f%5E%7B-1%7D%28y%29%3Dy%2B2&id=GWj6U), or 
%3Dx%2B2#card=math&code=f%5E%7B-1%7D%28x%29%3Dx%2B2&id=cnQ5v).
(2). 
.
.
%20%3D%202#card=math&code=%5Cln%207.389%20%5Capprox%20%5Cln%20%28e%5E2%29%20%3D%202&id=gzvPt)
‘%20%3D%203%5Ex%20%5Cln%20(3)#card=math&code=%283%5Ex%29%27%20%3D%203%5Ex%20%5Cln%20%283%29&id=rw0af). More generally 
%7D#card=math&code=a%5Ex%20%3D%20e%5E%7Bx%20%5Cln%20%28a%29%7D&id=EDy1t) .
Proof: 
‘%20%3D%20(e%5E%7Bx%20%5Cln%20(a)%7D)’%20%3D%20e%5E%7Bx%20%5Cln%20(a)%7D%20(x%20%5Cln%20(a))’%20%3D%20e%5E%7Bx%20%5Cln%20(a)%7D%20%5Cln%20(a)%20%3D%20a%5Ex%20%5Cln%20(a)#card=math&code=%28a%5Ex%29%27%20%3D%20%28e%5E%7Bx%20%5Cln%20%28a%29%7D%29%27%20%3D%20e%5E%7Bx%20%5Cln%20%28a%29%7D%20%28x%20%5Cln%20%28a%29%29%27%20%3D%20e%5E%7Bx%20%5Cln%20%28a%29%7D%20%5Cln%20%28a%29%20%3D%20a%5Ex%20%5Cln%20%28a%29&id=qt3bn) .
Remark: Because of the formulae 
‘%3De%5Ex#card=math&code=%28e%5Ex%29%27%3De%5Ex&id=Lg9Mr), 
, the natural logarithm is preferred in calculus.
(3) In calculus, all angles are measure in radians. Recall that 
.
Example: 
%3D%20%5Csin(%5Cpi)%3D0#card=math&code=%5Csin%20%28180%5E%5Ccirc%29%3D%20%5Csin%28%5Cpi%29%3D0&id=WSuTd).
%2C%20%5Ccos(x)%2C%20%5Ctan(x)%2C%20%5Ccot(x)#card=math&code=%5Csin%28x%29%2C%20%5Ccos%28x%29%2C%20%5Ctan%28x%29%2C%20%5Ccot%28x%29&id=c5iM2) 
etc. …
More Examples: 
%20%3D1%20%2C%20%5Ccos(%5Cpi%2F2)%3D0%2C%20%5Ctan(%5Cpi%2F4)%3D1%2C%20%5Ccot(%5Cpi%2F6)%3D%5Csqrt%7B3%7D%2C%20%5Ctan%5E2(%5Cpi%2F6)%3D%201%2F3%3D(%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D)%5E2#card=math&code=%5Csin%28%5Cpi%2F2%20%29%20%3D1%20%2C%20%5Ccos%28%5Cpi%2F2%29%3D0%2C%20%5Ctan%28%5Cpi%2F4%29%3D1%2C%20%5Ccot%28%5Cpi%2F6%29%3D%5Csqrt%7B3%7D%2C%20%5Ctan%5E2%28%5Cpi%2F6%29%3D%201%2F3%3D%28%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%29%5E2&id=hNbSl) .
%20%3D%5Csin%5E%7B-1%7D(1)%3D%20%5Cpi%2F2#card=math&code=%5Carcsin%281%29%20%3D%5Csin%5E%7B-1%7D%281%29%3D%20%5Cpi%2F2&id=VhbLa)
%3D%5Ccos%5E%7B-1%7D(0)%3D%5Cpi%2F2#card=math&code=%5Carccos%280%29%3D%5Ccos%5E%7B-1%7D%280%29%3D%5Cpi%2F2&id=WzoNM)
%20%3D%5Ctan%5E%7B-1%7D(1)%3D%5Cpi%2F4#card=math&code=%5Carctan%281%29%20%3D%5Ctan%5E%7B-1%7D%281%29%3D%5Cpi%2F4&id=PTEVR)
%3D%5Cpi%2F6#card=math&code=%5Ccot%5E%7B-1%7D%28%5Csqrt%7B3%7D%29%3D%5Cpi%2F6&id=EAbGL)
%3D%5Cpi%2F6#card=math&code=%5Carctan%281%2F%5Csqrt%7B3%7D%29%3D%5Cpi%2F6&id=lD0vn) . 
%20%5CRightarrow%20%5Csqrt%7By%7D%20%3D%20%5Ctan(x)%20%5CRightarrow%20x%20%3D%20%5Ctan%5E%7B-1%7D(%5Csqrt%7By%7D)#card=math&code=y%20%3D%20%5Ctan%5E2%28x%29%20%5CRightarrow%20%5Csqrt%7By%7D%20%3D%20%5Ctan%28x%29%20%5CRightarrow%20x%20%3D%20%5Ctan%5E%7B-1%7D%28%5Csqrt%7By%7D%29&id=T1KSg) So again we obtain
#card=math&code=%5Cpi%2F6%20%3D%20%5Ctan%5E%7B-1%7D%28%5Csqrt%7B1%2F3%7D%29&id=if3aY) .
Question: In general 
%3D1%2C%20%5Cforall%20n#card=math&code=%5Csin%28%5Cpi%2F2%2B2n%5Cpi%20%29%3D1%2C%20%5Cforall%20n&id=BpWG2) integral. Why not %3D%5Cpi%2F2%20%2B%202n%5Cpi#card=math&code=%5Carcsin%281%29%3D%5Cpi%2F2%20%2B%202n%5Cpi&id=AiYvk)?
Answer: This is because the range of the inverse sine function is set to be 
. See page 87 in the Openstax textbook.
