Trigonometry
Section 6.1 D1: Inverse Trig Functions
Name:
Introduction: With a calculator, give the degree measure of  .
1
1
1.   arcsin
2.   arcsin
2
2
WHY? Back to Algebra 2!
1. Which ones are functions? Why?
A
B
C
x
y
x
y
x
1
2
1
0
1
2
2
1
1
2
3
3
2
2
3
4
4
3
3
4
y
5
6
7
8
2. Switch the x and y coordinates from #1. Which ones are functions now? Reflecting across the line y=x is the inverse f
A
x
2
2
3
4
B
y
1
2
3
4
x
0
1
2
3
y
1
1
2
3
C
x
5
6
7
8
1
( x)
y
1
2
3
4
The last table C is the goal. We want to have a function that when we do the inverse(switch the x and y
coordinates), we get a function as well. To insure this, each y coordinate is used only once, we call this a oneto-one function.
Vertical line test-helps to make sure the first graph is a function.
Horizontal line test-helps to make sure the inverse is 1-1, which insures it is a function.
Examples from Algebra 2: Sketch: Vertical line test/Horizontal line test
f ( x)  2 x  4
f 1 ( x) =
BACK TO TRIGONOMETRY
1.   arcsin
1
2
  30
Let’s start with the
2.   arcsin
1
2
  30
y  sin x : Sketch
WAIT, it doesn’t pass the horizontal line test! Let’s put a restriction on it. Now it will pass the test! YES!!! So it does have an inverse.
y  sin x
Observations:
solve for y y  sin 1 x
x  sin y

y  sin 1 x or y  arcsin x : means the same :: Domain :  1,1 :: Range :  90 ,90

Quadrants I  positive Quadrant IV  negative
Answers between  90 ,  89 ,  88...88 , 89 , 90
WHY?
1.   arcsin
1
2
  30
2.   arcsin
1
2
  30
Can I write for problem 2 the answer   330 ? Does it have to be   30
Examples: Find the exact value.
1. y  sin 1 0
2. y  sin 1 1
5. y  sin 1  1
6. y  sin 1
Find the exact value.
Sketch y  cos x
1
2
3. y  sin 1
1
2
4. y  arcsin
7. y  sin 1
 3
2
8. y  sin 1
1
1.   arccos
2
2.   cos1
x  cos y
Inverse
3
2
 2
2
1
2
solve for y y  cos1 x
Observations:
y  cos1 x or y  arccosx : means the same :: Domain :  1,1 :: Range : 0 ,180
Quadrants I  positive Quadrant II  negative

Answers between 0 , 1 , 2...178 , 179 , 180
Examples: Find the exact value.
1. y  cos 1 0
2. y  cos1
1
2
3. y  cos1
 2
2

Tangent, Cotangent, Secant and Cosecant will be a found in the textbook. Here is an overview.
y  tan 1 x or y  arctan x
which means x  tan y
   
Domain :  ,   :: Range : 
,
 2 2 
Quadrants I and IV Quadrants of the unit circle from which range values come
y  cot 1 x or y  arc cot x
which means x  cot y
Domain :  ,   :: Range :  0,  
Quadrants I and II
Quadrants of the unit circle from which range values come
y  sec 1 x or y  arc sec x
Domain :  , 1
Quadrants I and II
    
Range : 0,   ,  
 2 2 
Quadrants of the unit circle from which range values come
1,   ::
y  csc 1 x or y  arc csc x
Domain :  , 1
Quadrants I and IV
which means x  sec y
which means x  csc y
     
Range : 
, 0   0,
 2   2 
Quadrants of the unit circle from which range values come
1,   ::
General statements about Inverses
1
1. f 1 ( x) 
f ( x)
2. The inverse function of the 1-1 function f is defined as f 1  ( y, x) | ( x, y) belongs to f 
3. The domain of f  range of f 1
The range of f  domain of f 1
4. For all x in the domain of f , f 1  f ( x)  x ,
for all x in the domain of f 1 , f  f 1 ( x)   x
Condensed Version:
sin 1 x
tan 1 x
csc1 x
are in quadrants I  positive and IV  negative
 90  90
cos1 x cot1 x
sec1 x
are in quadrants I  positve and II  negative
0  180
Using the calculator:
1
sec 1 x  cos 1  
x
1
csc 1 x  sin 1  
 x
1
cot 1 x  tan 1  
x
if
1
cot 1 x  tan 1    
x
if x  0
x0
1
NOTE: If in degrees, cot 1 x  tan 1    180  if x  0
 x
Examples: Find the exact real value.(radians)
1. y  tan 1  3
2. y  tan 1 0
Special Case
3. y  arc cot
3
3
5. y  arc csc 2
 
7. y  arc csc 2
4. y  arc cot
 3
3
6. y  arc sec 2
8. Give the exact value:
1

sin  arccos 
4
