16 – INVERSE
TRIGONOMETRIC FUNCTIONS
MATHEMATICS 201-NYA-05
PHILIP FOTH
1. Find the exact numerical value of each expression:


(a) arccos
3
2
 1 
(b) arcsin  

2

(c) arctan  3
(f) arccsc
2
3
3

(g) sin  arccos 
5



(h) arccot  tan 
5

2. Simplify the expression: (a)
cos (arccsc x)
(d) arccot 0
(e) arcsec 2
(i) csc(arctan 2)
x

(b) tan  arcsec 
3

5

(c) sin  2arccot 
x

3. Find the derivative of the given function and simplify your answer:
(a) y  arccos x
(b) f ( x)  x  arccsc x
 et  e  t 
(e) h(t )  arccot 

 2 
(h) w( z )  5arccot
z 1
(c) g ( x)  earcsin x
(f) y  arcsec 1  x 2
(i) H ( x)  arcsec x  arcsec
(d) k ( x) 
arctan x
x2
(g) f ( x)  arccos (1  2 x)
x
x2 1
x


(j) v( x)  ln 

2
 arccos x 
4. Find an equation of the tangent line to a graph of a given function at a given point:
(a) f ( x)  arctan
x
4
at x  4
 x 
(b) g( x)  arcsin 
 at x  1
 x 1 
5. Find an equation of the normal line to the curve g ( x)  arccsc
2
at the point x  3 .
x
ANSWERS
1. (a)
2. (a)

6
x2 1
x
3. (a) y  
(d) k ( x) 
(c) 
(b)
x2  9
3
1

3
5arccot z 1  ln 5
2 z z 1
5. y  2 3 x  6 3 

4
(e)

3

3
(g)
4
5
(h)
3
10
(i)
5
2
2
(e) h(t )  
1
x
2 3


6
x2 1
2
e  et
t
(j) v( x) 
(i) H ( x)  0
(b) y  
(f)
10 x
x  25
(c)
(b) f ( x)  arccsc x 
2 x  x2
x  2  4
8

2
(d)
x  2arctan x (1  x 2 )
(1  x 2 ) x3
(h) w( z )  
4. (a) y 

4
(b) 

1
2 3
(c) g ( x) 
(f) y 
earcsin x
1  x2
1
1  x2
(g) f ( x) 
1
2

x arccos x  1  x 2
1
x  x2