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```Lesson 4.7 Inverse trigonometric functions
I. What do we know about inverse functions?
II. Who are the inverse trigonometric
functions?
III. How do we evaluate inverse trigonometric
function without a calculator?
IV. How do we compose trigonometric
functions with inverse trig functions?
I. we know:
1.
2.
3.
4.
functions and their inverses are symmetric
by y = x line
their composition is equal to x f -1(f(x)) =
x, and f(f-1(x)) = x
domain of f(x) is the range of f-1 (x) and
range of f(x) is the domain of f-1 (x)
the function has to be one to one ( pass
the horizontal line test)
II. Trigonometric functions do not pass
Horizontal line test, so they do not have an
inverse unless we restrict the domain
1.graph sin(x) on window - л ≤ х ≤ л and л/2 ≤ у ≤ л/2 and see on what interval
the function pass the horizontal line test
2.graph the inverse of sine function: sin-1(x)
also called arcsine of x and sketch the
Sin (x)
domain
Sin -1(x) or
arcsin (x)
[- л/2,л/2] [-1 , 1 ]
Range
[-1 , 1 ]
[- л/2 ,л/2]
Graph cos(x) and cos -1 (x) find the restricted
domain and range and sketch the graph
Window – 1.3&lt; x &lt; л and – 1.3&lt; y &lt; л
cos (x)
cos -1(x) or
arccos (x)
domain [o, л]
[-1 , 1 ]
Range [-1 , 1 ] [o, л]
Graph tan(x) and tan-1(x) , find restricted domain
and sketch the graph, standard window.
tan (x)
tan -1(x) or
arctan (x)
All real
domain [- л/2,
л/2]
Range All real [- л/2 , л/2]
III how do I solve without calculator. (and with
calculator)
1.
sin – 1 (1/2 )
7.
cos – 1 (1 )
2.
sin

3


-1  

 2 
8.
cos – 1(cos(- 1.1 ))
3.
sin – 1 (л /2 )
9.
4.
sin – 1(sin л /9 )
10. sin( arctan (  2 ))
sin – 1(sin 5л / 6)
tan – 1 (  3 )
11. tan (arcsin (- 5/7 )
5.
6.
cos – 1

2


 2 


12. tan (cos -1 (0.28))
When you solve it is important to
remember where is the inverse function
interval on the unit circle
1. sin – 1 (1/2 ) = x then sin x = &frac12; that means
x = л/ 6 or 30&ordm; .
2. sin

3



-1 
=
 2 
x then sin x =

3


 2  and


x=
240&deg; or 300&deg; . we choose 300&deg; (- л /3)
because is in the interval (- л /2, л /2)
3. sin – 1 (л /2 ) no solution because domain of
arcsin is [ -1, 1] and л /2 &gt;1
4. sin – 1(sin л /9 ) remember composition of
function ? f-1(f(x) = x our functions are
inverses of each other so the answer is л /9
5. sin – 1(sin 5л / 6) this should be 5л / 6 but is
not because it is not in the restricted interval
of sin. So we choose the other quadrant
where sin has the same value л - 5л/6 =
л/6
6. tan – 1 (  3 ) = x then tan x = (  3 )
and x = - 60&deg; or – л/3
7. cos – 1 (1 ) = x then cos x = 1 and x is zero
8. cos – 1(cos(- 1.1 )) should be -1.1 however it
is not in the restricted interval for cosine so
we choose the other value 1.1
IV How do we compose trigonometric functions
with inverse trig functions?
Composing trig function with arctan
Consider a right triangle with one angle , the
opposite side x, and adjacent side = 1.
2
1

x
Hypotenuse should be
Then tan θ = x and tan -1(x) = θ
x
Sin (tan -1x) =
1 x2
1
Cos (tan -1x) =
1  x 2 and so on
```