inverse trig deriv notes by changcheng2

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```									AP Calculus
Inverse Trig Derivative Notes

y = sin –1 x or y = arcsin x  this statement wants to know what angle makes the answer =
x.     You can also write this   x = sin y.

y = sin –1 ½

However, the domain must be restricted for each trigonometric function to have an
inverse. The accepted way to restrict each follows:

domain             range
y = sin-1x      [-1, 1]             
 2 , 2 
        

y = cos-1x      [-1, 1]            [0, ]

y = tan-1x      (-, )             
 , 
 2 2

y = cot-1x      (-, )            [0, ]

y = sec-1x      |x| > 1         [0, ] y      
2

|x| > 1           
 ,  y  0
y = csc x  -1                    2 2

What quadrants should you use for each trig function?

sec-1 2 =                     tan-1 –1 =               sin 1
3
2
              
csc 1  2  

You may need to build a triangle to solve:
x
sin(tan-1 x) =                  cos(sin-1 x2) =                       tan(sec-1 2 )=
Derivatives of inverse trig functions:

u
1. y = sin-1 a  Find y'                                      re-write as x =
 build a triangle
 differentiate implicitly
 use the triangle to sub
out the 'trig of y' function

Or you can memorize the following:
function                 derivative
u
y = arcsin a
NOTE: Which ones are
u
y = arcos a                                                    negative? Are any similar?

u
y = arctan a

u
y = arccot a

u
y = arcsec a

u
y = arccsc a

Ex) y = arccos (3x)             Ex) y = tan-1 (ex)                  Ex) y = sec-1 x2

Ex) y = sin-1 (¼ x)                            Ex) y = arcsec e2x

Assign: 5-6: Worksheet

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