7-6 The Inverse Trigonometric Functions

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7-6 The Inverse Trigonometric Functions

• Objective To find values of the inverse
  trigonometric functions.

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The Inverse Trigonometric Function
When does a function have an inverse?
It means that the function is one-to-one. One-to-o
ne means that every x-value is assigned no more
than one y-value AND every y-value is assigned no
more than one x-value.
How do you determine if a function has an inverse?
Use the horizontal line test (HLT).
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The Inverse Trigonometric Function
Inverse Sine Function
Recall that for a function to have an inverse, it
must be a one-to-one function and pass the
Horizontal Line Test.
f(x) sin x does not pass the Horizontal Line
Test
and must be restricted to find its inverse.
sin x has an inverse function on this interval.
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The Inverse Trigonometric Function
The inverse sine function is defined by y
arcsin x if and only if sin y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arcsin x is 1, 1.
The range of y arcsin x is ?/2 , ?/2.
Example 1
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The Graph of Inverse Sine
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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 lt x lt ? / 2. Thus, y sin-1 x
means sin y x, where - ? /2 lt y lt ? /2
and 1 lt x lt 1. We read y sin-1 x as y
equals the inverse sine at x.
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Finding Exact Values of sin-1x

• Let ? sin-1 x.
• Rewrite step 1 as sin ? x.
• Use the exact values in the table to find the
  value of ? in -?/2 , ?/2 that satisfies sin ?
  x.

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Example

• Find the exact value of sin-1(1/2)

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The Inverse Trigonometric Function
The other inverse trig functions are generated by
using similar restrictions on the domain of the
trig function. Consider the cosine function
Inverse Cosine Function
f(x) cos x must be restricted to find its
inverse.
cos x has an inverse function on this interval.
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The Inverse Trigonometric Function
The inverse cosine function is defined by y
arccos x if and only if cos y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arccos x is 1, 1.
The range of y arccos x is 0 , ?.
Example 2
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The Graph of Inverse Cosine
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The Graph of Inverse Cosine
What is the relation between arcsin(x) and
arccos(x) ?
arccos(x) (-1)arcsin(x) ?/2
arcsin(x) arccos(x) ?/2
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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 lt x lt ?. Thus,
• y cos-1 x means cos y x,
• where 0 lt y lt ? and 1 lt x lt 1.

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Inverses of Sine and Cosine
Sin(x) Domain Range -1y1
Arccos(x) Domain -1x1 Range 0y
Cos(x) Domain 0x Range -1y1
Arcsin(x) Domain -1x1 Range
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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 lt ?
lt ?, whose cosine equals -?3 /2 Step 2
Rewrite ? cos-1 x as cos ? x. We obtain cos
? (-?3 /2)
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Example cont.
Find the exact value of cos-1 (-?3
/2) Solution 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 Trigonometric Function
The other trig functions require similar
restrictions on their domains in order to
generate an inverse.
Like the sine function, the domain of the section
of the tangent that generates the arctan is
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The Inverse Trigonometric Function
The inverse tangent function is defined by y
arctan x if and only if tan y x.
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The domain of y arctan x is (-?,?) .
The range of y arctan x is (?/2 , ?/2).
Example 3
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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 lt x lt ?/2. Thus,
• y tan-1 x means tan y x,
• where - ? /2 lt y lt ? /2 and ? lt x lt ?.

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Graph of Tangent of x with an unrestricted domain.
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Inverse Tangent of x
Domain
Range Real Numbers
-1
Tan x y means that tan y x and
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The Graph of Inverse Tangent and Cotangent
What is the relation between arctan(x) and
arcot(x) ?
arccot(x) (-1)arctan(x) ?/2
arctan(x) arccot(x) ?/2
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Tan can also be called arctan
-1
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Composition of Functions
Composition of Functions f(f 1(x)) x
and (f 1(f(x)) x.
Inverse Properties
If 1 ? x ? 1 and ?/2 ? y ? ?/2,
then sin(arcsin x) x and arcsin(sin y) y.
If 1 ? x ? 1 and 0 ? y ? ?, then cos(arccos
x) x and arccos(cos y) y.
If x is a real number and ?/2 lt y lt ?/2,
then tan(arctan x) x and arctan(tan y) y.
If x is a real number and 0 lt y lt ?,
then cot(arccot x) x and arccot(cot y) y.
Example 5 tan(arctan 4) 4
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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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Inverse Trigonometric Functions

• Examplesin(arcsin(-1)) -1arcsin(sin(pi/4))
  pi/4, (domain between -pi/2 and pi/2).Another
  exampleIf you have sin(x) 1if you take
  arcsin of both sidesarcsin(sin(x)) arcsin(
  1)You get x arcsin(1) pi/2In another
  word, "what x can I choose so that if I take its
  sine, I will get 1"

Consider tan(x) with domain Its inverse is
arctan(x) or tan (x) The relationship between
them is tan(tan (x)) x tan (tan(x)) x
-1
-1
-1
This is the same for other trigonometric
functions.
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Inverse Keys are on your calculator.
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The Inverse Function of Sine

• Basic idea To find sin-1(½), we ask "what angle
  has sine equal to ½?" The answer is 30. As a
  result we say sin-1(½) 30. In radians this is
  sin-1(½) p/6.
• More There are actually many angles that have
  sine equal to ½. We are really asking "what is
  the simplest, most basic angle that has sine
  equal to ½?" As before, the answer is 30. Thus
  sin-1(½) 30 or sin-1(½) p/6.
• Details What is sin-1(½)? Do we choose 210,
  30, 330 , or some other angle? The answer is
  30. With inverse sine, we select the angle on
  the right half of the unit circle having measure
  as close to zero as possible. Thus sin-1(½)
  30 or sin1(½) p/6.
• In other words, the range of sin-1 is restricted
  to 90, 90 or .
• Note arcsin refers to "arc sine", or the radian
  measure of the arc on a circle corresponding to a
  given value of sine.
• Technical note Since none of the six trig
  functions sine, cosine, tangent, cosecant,
  secant, and cotangent are one-to-one, their
  inverses are not functions. Each trig function
  can have its domain restricted, however, in order
  to make its inverse a function.

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Question??

• If you had sin(x) / sin²(x), you are left with
  1/sin(x), right?So how come when you have
  sin(x)arcsin(x) you aren't left with 1? Isn't
  arcsin(x) the same as writing sin (x)?
  Therefore writing sin(x)arcsin(x) would be like
  writing sin(x)/sin(x) 1 But it isn't.... why
  not?

-1
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The notation for inverse functions in just a
shorthand way of writing the inverse. The -1
looks like an exponent but it is not an exponent.
Answer

• Because sin is not 1/sin(x). The reciprocal of
  the sine function is the cosecant function.
• The arcsine function is the inverse function for
  the sine function on the interval
  So they cancel each other under the
  composition of functions.
• sin(arcsin(x)) arcsin(sin(x)) x

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Evaluate each expression without using a
calculator.
If x, the sinx -1 and Since
sin (-p/2) -1, then Sin (-1) (p/2)
Whose tangent is v3
Since tan p/3 v3 then Tan v3 p/3
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Find using the degree and radian mode on the
calculator
150
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Find the following
-1

• Find Sin (0.8) with a calculator.
• Degree mode 53
• Radian mode 0.93
• Find Cos (-0.5) without a calculator.
• Cos (-0.5) x means that cos x -0.5
  between 0 and p. Thus,
• Cos (-0.5) 2p/3

-1
-1
-1
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Find with and without a calculator.
Hypotenuse² will be (-2)² 3² v13 The cos is
adj/hyp 3/v13 Rationalize Denominator 3v13/13
v13
Calculator answer 0.83
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Find the approximate value (calculator) and exact
value (without a calculator)

• csc(cos (-0.4))

-1
5
-2
-0.4 in fraction form is -2/5 Cos adj/hyp Opp.
v 5² - (-2)² v21 Csc 1/sin hyp/opp
5/v21 Rationalize denominator
5v21/21 Calculator 1.09
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Example 8
a. sin1(sin (?/2)) ?/2
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Example 9
a. sin1(sin (3?/2)) ?/2
does not lie in the range of the arcsin
function, ?/2 ? y ? ?/2.
However, it is coterminal with
which does lie in the range of the arcsin
function.
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Example 10
Solution
3
2
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Finally, we encounter the composition of trig
functions with inverse trig functions. The
following are pretty straightforward
compositions. Try them yourself before you click
to the answer.
Did you suspect from the beginning that this was
the answer because that is the way inverse
functions are SUPPOSED to behave? If so, good
instincts but.
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Consider a slightly different setup
This is also the composition of two inverse
functions but
Did you suspect the answer was going to be 2?/3?
This problem behaved differently because the
first angle, 2?/3, was outside the range of the
arcsin. So use some caution when evaluating the
composition of inverse trig functions.
The remainder of this presentation consists of
practice problems, their answers and a few
complete solutions.
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Find the exact value of each expression without
using a calculator. When your answer is an angle,
express it in radians. Work out the answers
yourself before you click.
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On most calculators, you access the inverse trig
functions by using the 2nd function option on the
corresponding trig functions. The mode button
allows you to choose whether your work will be in
degrees or in radians. You have to stay on top of
this because the answer is not in a format that
tells you which mode you are in.
Use a calculator. For 21-24, express your answers
in radians rounded to the nearest hundredth.
Use a calculator. For 17-20, round to the nearest
tenth of a degree.
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Use a calculator. When your answer is an angle,
express it in radians rounded to the hundredths
place. When your answer is a ratio, round it to
four decimal places, but dont round off until
the very end of the problem.
Answers appear in the following slides.
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Answers for problems 1 9.
Negative ratios for arccos generate angles in
Quadrant II.
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Answers for 17 30.
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Assignment

• Page 289 2, 4, 5 8, 11 14
• Chapter 7 Test Wednesday