Download Angles, Degrees, and Special Triangles

Inverse Trigonometric Functions
Trigonometry
MATH 103
S. Rook
Overview
• Section 4.7 in the textbook:
– Review of inverse functions
– Inverse sine function
– Inverse cosine function
– Inverse tangent function
– Inverse trigonometric functions and right triangles
2
Review of Inverse Functions
Review of Inverse Functions
• Graphically, a function f has an inverse if it passes the
horizontal line test
f is said to be one-to-one
• Given a function f, let f-1 be the relation that results
when we swap the x and y coordinates for each point
in f
• If f and f-1 are inverses, their domains and ranges are
interchanged:
– i.e. the domain of f becomes the range of f-1 & the range
of f becomes the domain of f-1 and vice versa
4
Review of Inverse Functions
(Continued)
• None of the six trigonometric functions have inverses
as they are currently defined
– All fail the horizontal line test
– We will examine how to solve this problem soon
5
Inverse Sine Function
Inverse Sine Function
• As mentioned earlier,
y = sin x has no
inverse because it fails
the horizontal line test
• However, if we RESTRICT
the domain of y = sin x,
we can force it to be
one-to-one
– A common domain
restriction is    x  
2
2
– The restricted domain now passes the horizontal line
test
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Inverse Sine Function (Continued)
• The inverse function of y = sin x is y = sin-1 x
– Switch all (x, y) pairs in
the restricted domain
of y = sin x
– A COMMON MISTAKE is
to confuse the inverse
notation with the
reciprocal sin x  1
sin x
• To avoid confusion, y = sin-1 x is often written as
1
y = arcsin x
– Pronounced “arc sine”
– Be familiar with BOTH notations
8
Inverse Sine Function (Continued)
• For the restricted domain of y = sin x:
D: [-π⁄2, π⁄2]; R: [-1, 1]
• Then for y = arcsin x:
D: [-1, 1]; R: [- π⁄2, π⁄2]
– Recall that functions and
their inverses swap
domain and range
– This corresponds to angle in either QI or QIV
y = sin-1 x and y = arcsin x both mean x = sin y
– i.e. y is the angle in the interval [- π⁄2, π⁄2]
whose sine is x
9
Inverse Sine Function (Example)
Ex 1: Evaluate if possible without using a
calculator – leave the answer in radians:
 1 
a) sin 

 2
1
b) arcsin(-2)
10
Inverse Sine Function (Example)
Ex 2: Evaluate if possible using a calculator –
leave the answer in degrees: sin 1  0.4664
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Inverse Cosine Function
Inverse Cosine Function
• As mentioned earlier,
y = cos x has no
inverse because it fails
the horizontal line test
• However, if we RESTRICT
the domain of y = cos x,
we can force it to be
one-to-one
– A common domain
restriction is 0  x  
– The restricted domain now passes the horizontal line
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test
Inverse Cosine Function
(Continued)
• The inverse function of y = cos x is y = cos-1 x
– Switch all (x, y) pairs in
the restricted domain
of y = cos x
– To avoid confusion,
y = cos-1 x is often
written as y = arccos x
• Pronounced “arc
cosine”
• Be familiar with BOTH notations
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Inverse Cosine Function
(Continued)
• For the restricted domain of y = cos x:
D: [0, π]; R: [-1, 1]
• Then for y = arccos x:
D: [-1, 1]; R: [0, π]
– This corresponds to an
angle in either QI or QII
y = cos-1 x and
y = arccos x both mean x = cos y
– i.e. y is the angle in the interval [0, π]
whose cosine is x
15
Inverse Cosine Function (Example)
Ex 3: Evaluate if possible without using a
calculator – leave the answer in radians:
a) arccos(-3⁄2)
b) cos-1(1)
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Inverse Tangent Function
Inverse Tangent Function
• As mentioned earlier,
y = tan x has no
inverse because it fails
the horizontal line test
• However, if we RESTRICT
the domain of y = tan x,
we can force it to be
one-to-one
– A common domain
restriction is    x  
2
2
– The restricted domain now passes the horizontal line
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test
Inverse Tangent Function
(Continued)
• The inverse function of y = tan x is y = tan-1 x
– Switch all (x, y) pairs in
the restricted domain
of y = tan x
– To avoid confusion,
y = tan-1 x is often
written as y = arctan x
• Pronounced “arc
tangent”
• Be familiar with BOTH notations
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Inverse Tangent Function
(Continued)
• For the restricted domain of y = tan x:
D: [-π⁄2, π⁄2]; R: (-oo, +oo)
• Then for y = arctan x:
D: (-oo, +oo); R: [-π⁄2, π⁄2]
– This corresponds to an
angle in either QI or QIV
y = tan-1 x and y = arctan x
both mean x = tan y
– i.e. y is the angle in the interval [-π⁄2, π⁄2]
whose tangent is x
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Inverse Tangent Function
(Example)
Ex 4: Evaluate if possible without using a
calculator – leave the answer in radians:
arctan  1
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Inverse Trigonometric Functions
and Right Triangles
Taking the Inverse of a Function
• Recall what happens when we take the inverse
of a function:
f 1  f x   f  f 1 x   x
• e.g. Given x = 3, because y = ln x and ex are inverses:
ln e3  e ln3  3
– In other words, we get the original argument
PROVIDED that the argument lies in the domain
of the function AND its inverse
– This also applies to the trigonometric functions
and their inverse trigonometric functions
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Inverse Trigonometric Functions
and Right Triangles
• The same technique does not work when the
functions are NOT inverses
– E.g. tan(sin-1 x)
• Recall the meaning of sin-1 x
  sin 1 x  x  sin   sin   x
• i.e. the sine of what angle results in x
• With this information, we can construct a right
triangle using Definition II of the Trigonometric
functions
– We can use the right triangle to find tan sin 1 x   tan 
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Inverse Trigonometric Functions
and Right Triangles (Example)
Ex 5: Evaluate without using a calculator:
a)
 1 1 
sin  sin

2

c)
 1 3 
csc tan

4

b)
7 

cos 1  cos

6 

d)
 1 1 
sin  cos

5

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Inverse Trigonometric Functions
and Right Triangles (Example)
Ex 6: Write an equivalent expression that
involves x only – assume x is positive:
 1 1 
sec sin

x

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Summary
• After studying these slides, you should be able to:
– State whether or not an argument falls in the domain
of the inverse sine, inverse cosine, or inverse tangent
– Evaluate the inverse trigonometric functions both by
hand or by calculator
– Evaluate expressions using inverse trigonometric
functions
• Additional Practice
– See the list of suggested problems for 4.7
• Next lesson
– Proving Identities (Section 5.1)
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