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MTH 113
Section 4.7 Inverse Trig Functions
Review:
Only one to one functions have inverses; they must pass the
Only one‐to‐one functions have inverses; they must pass the horizontal line test as well as the vertical line test.
f ( x) = x 2
f ( x) = x 2 , x ≥ 0
f‐1 is the symbol for the inverse of f. f‐1 does not mean the reciprocal of f.
If the point (a,b) is on the graph of f, then (b,a) is on the graph of f‐1.
f and f‐1 are reflections of each other about the line y = x.
f and f
are reflections of each other about the line y = x
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A Function and Its Inverse
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Find the exact value of sin −1 0.
Find
i d the
h exact value
l off sin
i −1 − 1.
1
Find the exact value of sin −1
− 3
.
2
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Find the exact value of cos−1 − 1.
Find the exact value of cos −1
2
.
2
Find the exact value of cos −1
3
.
2
5
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Find the exact value of tan −1
3
.
3
Find the exact value of tan −1 − 1.
Find the exact value of tan −1 − 3.
Using a Calculator to Evaluate Inverse Trigonometric Functions
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Use a calculator to find each of the following in radians rounded to four decimal places.
sin‐1 0.47
tan‐1(‐30)
sin‐1 (‐0.625)
tan −1 − 5061
(
)
cos‐1 (4/9)
⎛ 7⎞
cos −1 ⎜⎜
⎟⎟
⎝ 10 ⎠
Inverse Properties
The Sine Function and its Inverse
sin(sin‐1x) = x for every x in [‐1, 1]
x) = x for every x in [‐1 1]
‐1
sin (sinx) = x for every x in ⎡⎢− π , π ⎤⎥
⎣ 2
2⎦
The Cosine Function and its Inverse
cos(cos‐1x) x) = x for every x in [
x for every x in [‐1,
1, 1]
1]
‐1
cos (cos x) = x for every x in [0, π]
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The Tangent Function and its Inverse
tan(tan‐1x) = x for every real number x
tan‐1(tan x) = x for every x in ⎛ − π , π ⎞
⎜
⎝ 2
⎟
2⎠
Find exact values without the use of a calculator.
cos(cos‐10.57)
2π ⎞
⎛
cos −1 ⎜ cos
⎟
3 ⎠
⎝
4π ⎞
⎛
cos −1 ⎜ cos
⎟
3 ⎠
⎝
4π
4π
is not in [ 0, π ] , so
is not the answer.
3
3
You must find what angle in [0, π ] has the same cosine as that of
4π
.
3
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tan(tan‐1 380)
⎡ ⎛ π⎞ ⎤
tan −1 ⎢ tan ⎜ - ⎟ ⎥
⎣⎢ ⎝ 3 ⎠ ⎥⎦
⎡
3π ⎤
tan −1 ⎢ tan
⎥
4 ⎦
⎣
cos‐1(cos 2π)
cos(cos‐13π)
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Use a sketch to find exact values.
sin(tan‐1 7/24)
cos[sin‐1(‐4/5)
tan[cos‐1(‐ ¼)]
⎡
⎛
2 ⎞⎤
sec ⎢sin -1 ⎜⎜ −
⎟⎟ ⎥
⎢⎣
⎝ 2 ⎠ ⎥⎦
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