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Inverse Trigonometric Functions
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Definition of the Inverse Function
•
Let f and g be two functions such that f (g(x)) = x for
every x in the domain of g and g(f (x)) = x for every x in
the domain of f.
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The function g is the inverse of the function f, and
denoted by f -1 (read “f-inverse”).
•
•
Thus, f ( f -1(x)) = x and f -1( f (x)) = x.
The domain of f is equal to the range of f -1, and vice
versa.
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Text Example
Show that each function is the inverse of the other:
f (x) = 5x and g(x) = x/5.
Solution
To show that f and g are inverses of each other, we must show that f (g(x)) = x
and g( f (x)) = x. We begin with f (g(x)).
f (x) = 5x
f (g(x)) = 5g(x) = 5(x/5) = x.
Next, we find g(f (x)).
g(x) = 5/x
g(f (x)) = f (x)/5 = 5x/5 = x.
Notice how f -1 undoes the change produced by f.
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Finding the Inverse of a Function
The equation for the inverse of a function f can be found as
follows:
1.
2.
3.
4.
Replace f (x) by y in the equation for f (x).
Interchange x and y.
Solve for y. If this equation does not define y as a function
of x, the function f does not have an inverse function and
this procedure ends. If this equation does define y as a
function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 with f -1(x).
We can verify our result by showing that f ( f -1(x)) = x and
f -1( f (x)) = x.
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Text Example
Find the inverse of f (x) = 7x – 5.
Solution
Step 1
Replace f (x) by y.
y = 7x – 5
Step 2
Step 3
Step 4
Interchange x and y.
x = 7y – 5
This is the inverse function.
x + 5 = 7y
x+5=y
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Add 5 to both sides.
Solve for y.
Replace y by f -1(x).
f -1(x) =
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x+5
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Divide both sides by 7.
Rename the function f -1(x).
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The Horizontal Line Test For Inverse Functions

A function f has an inverse that is a function, f –1, if
there is no horizontal line that intersects the graph
of the function f at more than one point.
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The Inverse Sine Function
The inverse sine function, denoted by sin-1, is the inverse of the restricted sine
function y = sin x, -  /2 < x <  / 2. Thus,
y = sin-1 x means sin y = x,
where -  /2 < y <  /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the
inverse sine at x.”
y
y = sin x
-  /2 < x < /2
1
-  /2
x
 /2
-1
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Domain: [-  /2,  /2]
Range: [-1, 1]
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The Inverse Sine Function
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Finding Exact Values of sin-1x
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Example

Find the exact value of sin-1(1/2)
  sin
1
1
2
1
sin  
2
 1
sin

6 2
 
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
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The Inverse Cosine Function
The inverse cosine function, denoted by cos-1, is the
inverse of the restricted cosine function
y = cos x, 0 < x < .
Thus, y = cos-1 x means cos y = x,
where 0 < y <  and –1 < x < 1.
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Text Example
Find the exact value of cos-1 (-3 /2)
Solution Step 1
Let  = cos-1 x. Thus  = cos-1 (-3 /2)
We must find the angle , 0 <  < , whose cosine equals -3 /2
Step 2
Rewrite  = cos-1 x as cos  = x. We obtain cos  = (-3 /2)
Step 3
Use the exact values in the table to find the value of  in [0, ]
that satisfies cos  = x. The table on the previous slide shows that the
only angle in the interval [0, ] that satisfies cos = (-3 /2) is 5/6.
Thus,  = 5/6
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The Inverse Tangent Function
The inverse tangent function, denoted by tan-1, is the
inverse of the restricted tangent function
y = tan x, -/2 < x < /2.
Thus, y = tan-1 x means tan y = x,
where -  /2 < y <  /2 and –  < x < .
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Inverse Properties
The Sine Function and Its Inverse
sin (sin-1 x) = x for every x in the interval [-1, 1].
sin-1(sin x) = x
for every x in the interval [-/2,/2].
The Cosine Function and Its Inverse
cos (cos-1 x) = x for every x in the interval [-1, 1].
cos-1(cos x) = x for every x in the interval [0, ].
The Tangent Function and Its Inverse
tan (tan-1 x) = x for every real number x
tan-1(tan x) = x for every x in the interval (-/2,/2).
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