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Inverse Trig Functions.notebook
November 12, 2013
Pre-Calculus
Chapter 7
Inverse Trig Functions
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Inverse Trig Functions.notebook
November 12, 2013
Objective:
• Be able to evaluate inverse trigonometric funcons
• Be able to explain the differences between a trig funcon and its inverse
• Be able to write equaons for inverses of trigonometric funcons
• Graph inverses of trigonometric funcons
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Pre­Calculus
Graphing Inverses of Trig functions
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Inverse Trig Functions: Cos­1 (x) takes a value x and produces an angle y
and Cos(x) and 1/Cos(x) =sec(x) takes an angle x and
produces a number value y
so these functions use two different types of x's
therefore their y's will never equal each other
Inverse functions undo the functions so we can get the
input value.
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Inverse Trig Functions
sin(x) = y cos(x) = y tan(x) = y x = sin‐1 (y) or arcsin(y) = x
x = cos‐1 (y) or arccos(y) = x
x = tan‐1 (y) or arctan(y) = x
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Examples
A = cos(b)
cos‐1 (A) = b or arccos(A) = b
Tan(θ) = z
tan‐1 (z) = θ or arctan(z)= θ
cot(m) = p
cot‐1 (p) = m or arccot(p)= m
c = csc(d)
csc‐1 (c) = d or arccsc(c) = d
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ex. Find all posive values of x for which cosx = ½ look on your unit circle
x = cos‐1 (1/2) so x = 60, 300, 420….. and π/3, 5 π/3, 7 π/3……
so between 0 and 360, we have 60 and 300
ex. Find all posive values of x for which sin x = (1/ √2) between 0 and 360
Sinx = so 45, 135
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Ex. Evaluate each expression. Assume that all angles are in quad 1
cos(arccos (½)) = ½
cos(arccsc(5/3)) = 4/5
sec(arctan (7/13)) = √218/13
On your own: sin(arcsin(13/14) = 13/14
cot(tan‐1(5/10) = 10/5
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Ex. Verify each equaon.
arccos + arcsin = arctan 1 + arccot 1
tan‐1 (3/4) + tan‐1 (5/12) = tan‐1 (56/33)
On your own:
arcsin (3/5) + arcos (15/17) = arctan (77/36)
tan‐1 1 + cos‐1 = sin‐1 1/2 + sec‐1 √2
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sin(x)
Domain (all reals)
Range(‐1<y<1)
(x, sin(x))
(90 o, 1)
Funcon
sin(x) = (1/2)
The value of sine of an angle x is 1/2
sin‐1 (x) or arcsin(x)
Domain [‐1,1]
Range (all reals)
(sin(x), x)
(1, 90 o)
not a funcon (failed vercal line test)
x = sin‐1(1/2)
x is all the angles whose sine is 1/2, x = 30o, 150,and so on
Sin(x)
Domain [‐90,90]
Range[‐1,1]
(x, sin(x))
(90 o, 1)
Funcon
sin(x) = (1/2)
The value of sine of an angle x is 1/2
Sin‐1 (x) or Arcsin(x)
Domain [‐1,1]
Range [‐90,90]
(sin(x), x)
(1, 90 o)
not a funcon (failed vercal line test)
x = sin‐1(1/2)
x is only the angles whose sine is 1/2 and between ‐90 o and 90 o so only 30 o
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cos(x)
Domain (all reals)
Range(‐1<y<1)
(x, cos(x))
(90 o, 0)
Funcon
cos(x) = (1/2)
The value of cosine of an angle x is 1/2
cos‐1 (x) or arccos(x)
Domain [‐1,1]
Range (all reals)
(cos(x), x)
(0, 90 o)
not a funcon (failed vercal line test)
x = cos‐1(1/2)
x is all the angles whose cosine is 1/2, x=60 o,120 o, and so on
Cos(x)
Domain [0,180]
Range[0,1]
(x, cos(x))
(90 o, 0)
Funcon
cos(x) = (1/2)
The value of cosine of an angle x is 1/2
Cos‐1 (x) or Arccos(x)
Domain [‐1,1]
Range [0,180]
(cos(x), x)
(0, 90 o)
Funcon! (now it passes vercal line test)
x = Cos‐1(1/2)
x is only the angles whose cosine is 1/2 and between 0 o and 180 o which would be only 60 o
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tan(x) = y tan‐1x=y or
arctan(x) = y
X
Y
X
Y
Tan‐1x=y or
Arctan(x) = y
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cot(x) = y cot‐1x=y or
arccot(x) = y
X
Y
X
Y
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Cot(x) = y Cot‐1(x) = y or
X
Y
X
Y
Arccot(x) = y
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sec(x) = y
sec‐1(x) = y or X
Y
X
Y
arcsec(x) = y
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Sec(x) = y
X
Y
X
Y
Sec‐1x = y or Arcsec(x) = y
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csc(x) = y
X
Y
X
Y
csc‐1(x) = y or arccsc(x) = y
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Csc(x) = y
X
Y
X
Y
Csc‐1(x) = y or
Arccsc(x) = y
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1) What is the domain of the funcon 2)write the inverse funcon (Switch x and y, solve for y)
Y = Arctan(x) : Y = Sin(x) ‐ 45 :
Y = Arctan (2x)
:
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Determine if each of the following is true or false. If false, give a counterexample.
Make table: x, Inside parenthesis, outside parenthesis; see if last matches first
X Tan­1(x) tan(Tan­1x)
tan(Tan‐1(x)) = x for all x
1
2
3
Cot‐1(cot(x)) = x for all x
Cos‐1 (x) = 1/Cos(x) 45
63.4
71.5
1
2
3
X cot(x) Cot­1(cot(x))
45
1
45
90
0
90
135 ­1
­45
X Cos­1(x) 1/Cos(x)
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Evaluaon:
• True or False: if sin(x) = y then y is a number value between ‐1 and 1
• True or False: if arcsin(x) = y then y is an angle without a range
• True or False: Arcsin(x)=y is exactly the same as arcsin(x)=y except for the fact that arcsin(x) has a restricted domain.
• True or False: The domain of Tan‐1 (x) =y is restricted
• True or False: The restricted range of Arccos(x)=y is the restricted domain of Cos(x)=y
Practice:
Finish pg. 429 Do pg. 452 #29­31,35,36, finish Review Packet and worksheet
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Homework:
pg. 331 16 – 44 evens
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