PC 4.7 Notes – Inverse Trig Functions
1
PC 4.7 Notes – Inverse Trig Functions

y  sin x
4
4
2
2
-5
-5
5
5
-2
-2
-4
-4
y  sin x does NOT pass HLT for inverses, so restrict to interval 
D: [
 
No longer periodic
, ]
2 2
R:  1  y  1
Take inverse: switch axes, switch D and R


to
2
2
D:
R:
Notation: sin y  x  y  sin 1 ( x) or y  arcsin x
(LOOKING FOR ANGLE MEASURE)
1
EX 1: a) arcsin( )
2

b) sin 1 (
3
)
2
c) sin 1 2
y  cos x
4
4
2
2
-5
-5
5
5
-2
-2
-4
-4
y  cos x does NOT pass HLT for inverses, so restrict to interval 0 to 
D: [0,  ]
No longer periodic
D:
R:  1  y  1
R:
Take inverse: switch axes, switch D and R
Notation: cos y  x  y  cos 1 ( x) or y  arccosx
2
PC 4.7 Notes – Inverse Trig Functions

y  tan x

New D:
restrict
D: [

 
2, 2
] R: (-∞, ∞)
R:
1
EX 2: a) arccos( )
2
b) tan 1 (1)
Inverse trig functions are positive in ONLY one quadrant and negative in ONLY one
quadrant due to restricted domains.
cos1  (-)
sin 1  (+)
cos1  (+)
tan 1  (+)
sin 1  (-)
tan 1  (-)
Determining Inverse Trig Values Exactly
Find the exact value of each, if possible. Draw a reference triangle and solve for the
missing side. Use the chart above to remind you in which quadrants the functions are
defined.
EX 3:
 1 
 1 
1
a) arcsin  
b) cos1 
c) arctan  


2
3
 2

Estimating Inverse Trig Values with the Calculator
Check the MODE on your calculator.
1
Remember sin 1 
(inverse  reciprocal)
sin
EX 4: a) cos 1 (0.75 )
b) arcsin(0.99)
3
c) arctan(1.25)
PC 4.7 Notes – Inverse Trig Functions
Review Inverse Function
y  arcsin x
sin y  x
y  arccosx
y  arctan x
cos y  x
tan y  x
Domain
1  x  1
Range


 y
2
2
0 y 
1  x  1
  x  


2
 y
EX 5: Use an inverse trig function to write  as a function of x .
Inverse Properties
1. sin(arcsinx)  x and arcsin(sin y)  y if
 1  x  1 and 

2
 y

(-1.57 to 1.57)
2
2. cos(arccosx)  x and arccos(cosy)  y if
(0 to 3.14)
 1  x  1 and 0  y  
3. tan(arctanx)  x and arctan(tan y)  y if
x is any real number and 

2
3
3
)
2
2
Use the inverse property to simplify.

EX 6: arcsin(sin
a) tan(arctan(5))
 y

2
(-1.57 to 1.57)
(outside interval for y)
b) arcsin(sin
5
)
3
4

c) cos(cos1  )

2
PC 4.7 Notes – Inverse Trig Functions
Compositions of Functions –evaluate exactly (Sketch reference triangle)
1
3
)
EX 7: a) cos(arcsin
b) tan(sin 1 
( ))
5
2
3
c) sin(arcsin )
5
d) cos(tan1 
5
)
12
e) tan(arcos ½))
More challenging problems:
f) sin(arccos3x)
g) sin(arctan( x  1))
5