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Math 151 Fall 2016 Yi-Ching Wang
Section 4.6: Inverse Trigonometric Functions
Inverse sine function
arcsin x = sin−1 x = y ⇔ sin y = x
Cancellation Equations:
π
π
≤x≤
2
2
−1
sin(sin x) = x for − 1 ≤ x ≤ 1
sin−1 (sin x) = x for −
Inverse cosine function
arccos x = cos−1 x = y ⇔ cos y = x
Cancellation Equations:
cos−1 (cos x) = x for 0 ≤ x ≤ π
cos(cos−1 x) = x for − 1 ≤ x ≤ 1
Inverse tangent function
arctan x = tan−1 x = y ⇔ tan y = x
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Math 151 Fall 2016 Yi-Ching Wang
Cancellation Equations:
π
π
≤x≤
2
2
−1
tan(tan x) = x for − ∞ ≤ x ≤ ∞
tan−1 (tan x) = x for −
Inverse cotangent function
arccotx = cot−1 x = y ⇔ cot y = x
Inverse secant function
arcsecx = sec−1 x = y ⇔ sec y = x
Inverse cosecant function
arccscx = csc−1 x = y ⇔ csc y = x
Math 151 Fall 2016 Yi-Ching Wang
Example: Find the exact value of the expression.
√ !
3
=
(1) sin−1
2
(2) sin
−1
−1
2
(4) cos
=
√ !
2
=
2
(3) cos−1
−1
−1
2
=
5π
(5) arcsin sin
4
=
5π
(6) arccos cos
=
4
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Math 151 Fall 2016 Yi-Ching Wang
7π
(7) arctan tan
=
4
√ (8) sin arcsin 3 =
Example: Find the exact value of the expression.
1
−1
(1) cos sin
=
2
2
(2) sec tan−1
=
3
(3) sin (tan−1 x) =
(4) cos sin−1 x =
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Math 151 Fall 2016 Yi-Ching Wang
Derivatives of Inverse Trigonometric Functions
d
1
sin−1 (x) = √
dx
1 − x2
d
1
cos−1 (x) = − √
dx
1 − x2
1
d
tan−1 (x) =
dx
1 + x2
d
1
csc−1 (x) = − √
dx
x x2 − 1
d
1
sec−1 (x) = √
dx
x x2 − 1
d
1
cot−1 (x) = −
dx
1 + x2
Example: Find the derivative.
2
(1) y = sin−1 (2x)
(2) y = cos−1 (4x2 )
(3) y = arctan(sin(4x))
−1 (x)
(4) y = xsec
0
Reference: Math 151 lecture notes from Joe Kahlig and Mariya Vorobets.