slide 1-1 6 inverse trigonometric functions y. ath
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Slide 1-1
6Inverse Trigonometric Functions
Y. Ath
Slide 1-2
6.1 Inverse Circular Functions
6.2 Trigonometric Equations I
6.3 Trigonometric Equations II
6.4 Equations Involving Inverse Trigonometric Functions
Slide 1-3
Vertical Line Test
Horizontal Line Test
If a function f is one-to-one on its domain, then f has an inverse function
Slide 1-4
Inverse Function
The inverse function of the one-to-one function f is defined as follows.
Slide 1-5
Caution
The –1 in f –1 is not an exponent.
1 1( )
( )f x
f x
Slide 1-6
xxyy
xxyy
xxyy
tantan ,tan tan(3)
coscos ,coscos (2)
sinsin ,sinsin (1)
:Example
11-
11-
11-
yyff
xxff
)(
)(1
1
Slide 1-7
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.
11 :Range22
:Domain
y
x
Slide 1-8
2,
2Range ,11Domain ,sin
1,1Range ,2
,2
Domain ,sin
1
,-xy
yx
12
71.02
2
4
00
71.02
2
4
12
RangeDomain
6.12
1
8.04
71.0
00
8.04
71.0
6.12
1
RangeDomain
-1.5 -1 -0.5 0 0.5 1 1.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Slide 1-9
Slide 1-10
Find y in each equation.
Example FINDING INVERSE SINE VALUES
Slide 1-11
Example FINDING INVERSE SINE VALUES (cont.)
Slide 1-12
Example FINDING INVERSE SINE VALUES (cont.)
–2 is not in the domain of the inverse sine function, [–1, 1], so does not exist.
,2sin y 11 y
Slide 1-13
Inverse Cosine Function
Cos x has an inverse function on this interval.
f(x) = cos x must be restricted to find its inverse.
y
2
1
1
x
y = cos x
Slide 1-14
,0Range ,11Domain ,cos
1,1Range ,,0Domain ,cos1
,-xy
yx
1
71.02
2
4
3
02
71.02
2
4
10
RangeDomain
14.31
4.24
371.0
6.12
0
8.04
71.0
01
RangeDomain
-1.5 -1 -0.5 0 0.5 1 1.50
0.5
1
1.5
2
2.5
3
3.5
Slide 1-15
Inverse Tangent Functionf(x) = tan x must be restricted to find its inverse.
Tan x has an inverse function on this interval.
y
x
2
3
2
32
2
y = tan x
Slide 1-16
The inverse tangent function is defined by
y = arctan x if and only if tan y = x. The domain of y = arctan x is .( , )
The range of y = arctan x is [–/2 , /2].
Slide 1-17
Graphing Utility: Graph the following inverse functions.
a. y = arcsin x
b. y = arccos x
c. y = arctan x
–1.5 1.5
–
–1.5 1.5
2
–
–3 3
–
Set calculator to radian mode.
Slide 1-18
Graphing Utility: Approximate the value of each expression.
a. cos–1 0.75 b. arcsin 0.19
c. arctan 1.32 d. arcsin 2.5
Set calculator to radian mode.
Slide 1-19
Example:
a. sin–1(sin (–/2)) = –/2
1 5b. sin sin3
53 does not lie in the range of the arcsine function, –/2 y
/2. y
x
53
3
5 23 3 However, it is coterminal with
which does lie in the
range of the arcsine function.
1 15sin sin sin sin3 3 3
Slide 1-20
Example:
2Find the exact value of tan arccos .3
x
y
3
2
adj2 2Let = arccos , then cos .3 hyp 3
u u
2 23 2 5
opp 52tan arccos tan3 adj 2
u
u
Slide 1-21
Inverse Function Values
Slide 1-22
Trigonometric Equations I6.2Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities
Slide 1-23
Example
12sin 1, sin
2
Slide 1-24
Example 1(b)
SOLVING A TRIGONOMETRIC EQUATION BY LINEAR METHODS
Solve the equation 2 sinθ + 1 = 0 for all solutions.
210 360 , 330 360 ,
where is any integer
n n
n
Slide 1-25
Example SOLVING A TRIGONOMETRIC EQUATION BY FACTORING
Subtract sin θ.
Factor out sin θ.
Zero-factor property
Solution set: {0°, 45°, 180°, 225°}
Slide 1-26
Trigonometric Equations II6.3Equations with Half-Angles ▪ Equations with Multiple Angles
Slide 1-27
Example (a) over the interval
and
(b) for all solutions.
Slide 1-28
In-class exercises (pp 270-271)
Solution set: {30°, 60°, 210°, 240°}
Solution set, where 180º represents the period of sin2θ:
{30° + 180°n, 60° + 180°n, where n is any integer}
(1)
(2)
(3)
Solve tan 3x + sec 3x = 2 over the interval (4)Solution set: {0.2145, 2.3089, 4.4033}
Slide 1-29
Equations Involving Inverse Trigonometric Functions 6.4Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations
Slide 1-30
Example
Slide 1-31
Example:
Slide 1-32
Example
Slide 1-33
In class exercise
2
2 :Ans arcsinarccos Solve )3
3
32 :Ans 2cscsec Solve )2
3 :Ans arctan2 Solve 1)
11
xx
x
x
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