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Pre-Calculus Notes Section 6-8: Inverse Trigonometric Functions
The inverse of sin x is arcsin x or sin 1 x
If sin   .5 , then   sin 1 (.5)
The inverse of cos x is arccos x or cos 1 x
The inverse of tan x is arctan x or tan 1 x
All the inverses of trig functions are symbols for angles, just as  is a symbol for
angle. The symbol  A means “the angle whose vertex is point A.” Arcsin x means “
the angle whose sine is x.”
Evaluating Trig Functions
1. Graph 30 on a coordinate grid(just like at the beginning of the chapter), make a
triangle, and use your triangle knowledge to find all the sides of the triangle. Find
cos 30 , sin 30 , tan 30 .
2. Create three more coordinate grids. Graph 150 , 210 , 330 on their own grid.
Make triangles and find the lengths of all their sides.
a. What do you notice about the reference angles for each triangle?
b. Find cos, sin, tan of the reference angle for each triangle. This is also the cos, sin,
tan values for the original angles you graphed.
c. What do you notice about the cos, sin, tan values for 30 , 150 , 210 , 330 ?
d. Which cos values are the same? Which sin values are the same? Which tan
values are the same?
3. Create 4 more grids. Graph 45 , 135 , 225 , 315 on their own grid. Make triangles
and find the lengths of all their sides.
a. What do you notice about the reference angles for each triangle?
b. Find cos, sin, tan of the reference angle for each triangle. This is also the cos, sin,
tan values for the original angles you graphed.
c. What do you notice about the cos, sin, tan values for 45 , 135 , 225 , 315 ?
d. Which cos values are the same? Which sin values are the same? Which tan
values are the same?
4. Use your observations from 2 and 3 to answer the following questions.
a. Which quadrants will have the same cosine values? Why?
b. Which quadrants will have the same sine values? Why?
c. Which quadrants will have the same tangent values? Why?
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d. If you were told that cos   , how many  values could give you that
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cosine value? Which two quadrants could  come from?
e. How could you find a  value? How could you find the other value?
3
f. If you were told that sin  
, how many  values could give you that sine
5
value? Which two quadrants could  come from?
g. How could you find a  value? How could you find the other value?
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h. If you were told that tan   , how many  values could give you that
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tangent value? Which two quadrants could  come from?
i. How could you find a  value? How could you find the other value?
Conclusion
 In general, if you are given a sine, cosine, tangent value how many  angles
could have given you that value?
 State the rules to find  angles for a given sine, cosine, tangent value.
Trig function
Sine
Positive value
Negative value
Cosine
Tangent
Examples: Find the values for  that satisfy each equation.
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1.  = arccos 1
2. arccos
=
3. arcsin –1 = 
2
4. arcsin ½ = 
5. arctan 1 = 
6. arccot 1 = 
The graph of the sine function is the set of all ordered pairs ( , sin  ) . Therefore, the
graph of the inverse relation arcsin  is the set of all ordered pairs (sin  , ) . Meaning,
to graph the arcsin relation first graph sin  then switch the x and y coordinates.
1. Graph y  sin  from  360    360 .
2. Use the graph of y  sin  to sketch the graph y  arcsin  .
3. What are the domain and range for these two graphs?
4. Is y  arcsin  a function? Why or why not?
5. Do questions 1-4 for the graphs of cos and arccos  360    360 ,
and tan and arctan  180    180 .
Are the inverses of trig functions functions themselves?
Look at the graphs of your inverse relations again. Are there any consecutive  values
you could restrict your range to that would make the inverse relations functions? What
are they?
These values are called the principal values for the trig functions. When you are asked
to find  values that are prinicipal values for the functions, the trig functions will be
denoted by a capital first letter. Sin, Cos, Tan, Arcsin, Arccos, Arctan.
1
  .   60 . It does not equal 60 and 300 because 300 is
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not between 0  180 .
Example: For Arc cos
Trig Function
Principal Values
Sine
 90    90
Cosine
0    180
Tangent
 90    90
Principle Values without
negative angles
2. Which of the basic trig functions can have a y value of ½?
3. Which of the basic trig functions can have a y value of 2?
4. What is the domain and range for all 6 basic trig functions?
Solve for  without using a calculator.
3
1
5.   arctan( 1)
6.   arccos( )
7.   arcsin(  )
2
2
8.   arcsin( 
2
)
2
9.   arc cot(
3
)
3
10.   sec 1 (2)
3
)
2
11.   csc 1 (2)
12.   csc 1 (1)
13.   Sin 1 (
14.   Arc tan(1)
15.   Sin 1 (0)
1
16.   Sin 1 ( )
2
18.   Arc cos(0)
4
19.   cos(Cos 1 ( ))
5
3
20.   tan( Cos 1 ( ))
5
21.   cos( 2Tan 1 3 )
22.   Cos 1 (

24.   Sin 1 (cos )
2
25.   sin( Sin 1 (
17.   Tan 1 (
3
)
3
3
))
2
3
)
2
1
26.   sin( 2 Sin 1 ( ))
2
1
27.   cos(Tan 1 3  Sin 1 ( ))
2
1
28.   sin( Tan 11  Sin 1 )
2

2 
 )
29.   cos(Cos 1  
 2
2


 1
30.   cos( Sin 1     Cos 1 2)
 2