# Inverse Trigonometric Functions Differentiation

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```					Inverse Trigonometric Functions:
Differentiation
Interactive Trigonometry:
• http://colalg.math.csusb.edu/~devel/IT/

math100/notes/zoo/invtrig.html

• http://www.math.ucdavis.edu/~kouba/Cal
cOneDIRECTORY/invtrigderivdirectory/Inv
TrigDeriv.html
Inverse Trigonometric Functions Definition:
For x in the   the inverse trig    is the angle     whose trig
interval:        function:       measure in the   function is x.
interval:

[-1 , 1]         sin-1(x)        [-/2 , /2]     sin  = x
[-1 , 1]        cos-1(x)            [0, ]        cos  = x
(-, )         tan-1(x)         [-/2 , /2]     tan  = x
(-, )         cot-1(x)            [0, ]        cot  = x
(- , -1] or      sec-1(x)         [0, /2) or      sec  = x
[1, )                            (/2, ]
(- , -1] or       csc-1(x)        [- /2, 0) or    csc  = x
[1, )                             (0, /2]
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
Inverse Cotangent Function
Inverse Secant Function
Inverse Cosecant Function
Evaluating Expressions:
1. arccos(-1/2)
2/3
2. arcsin (0)
0
3. arccot (-3)
5/6
4. arccos (-0.8923)
5. arctan (-3)
Inverse Trigonometric Functions Properties:
• If -1  x  1 and -/2  y  /2, then:
–   sin(arcsin x) = x                and         arcsin(sin y) = y

• If -1  x  1 and 0  y  , then:
–   cos(arccos x) = x                and         arccos(cos y) = y

• If -  x   and -/2  y  /2, then:
–   tan(arctan x) = x                and         arctan(tan y) = y

•     Similar properties hold true for the other inverse trigonometric functions
Evaluating Expressions & Solving Equations:
6. tan(arccos(2/2))
1
7. sec(arctan(-3/5))
34/5
8. arctan(2x – 3) = /4
x=2
9. arctan(2x – 5) = -1
1.721
10. arccos(x) = arcsec(x)
x = 1
Inverse Trigonometric Functions Derivatives:

d                u'
[arcsin u] 
dx              1 u 2

d                u'
[arccos u] 
dx              1 u 2

d                u'
[arctan u] 
dx              1 u 2
Inverse Trigonometric Functions Derivatives:

d      1      u'
[cot u] 
dx           1 u  2

d      1      u'
[sec u] 
dx           u u2  1

d      1       u'
[csc u] 
dx           u u2  1

```
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