Section 4.6: Inverse Trigonometric Functions
y = arcsin x = sin−1 (x)
y = arccos x = cos−1 (x)
y = arctan x = tan−1 (x)
Example. Find the exact value of the expression.
√ !
3
−1
=
(A) sin
2
(B) sin
−1
1
−
2
=
M151 Notes
(C)
©, R.G. Lynch, Texas A&M
√ !
2
=
2
cos−1
(D) cos−1
−1
2
=
5π
(E) arcsin sin
4
5π
(F) arccos cos
4
=
7π
(G) arctan tan
4
(H) sin arcsin
=
√ 3 =
=
Section 4.6: Inverse Trigonometric Functions Page 2 of 5
M151 Notes
©, R.G. Lynch, Texas A&M
Example. Find the exact value of the expression.
−1 1
=
(A) cos sin
2
2
−1
(B) sec tan
=
3
(C) sin(tan−1 (x) =
(D) cos sin−1 x =
Section 4.6: Inverse Trigonometric Functions Page 3 of 5
M151 Notes
©, R.G. Lynch, Texas A&M
Section 4.6: Inverse Trigonometric Functions Page 4 of 5
Derivatives of Inverse Trigonometric Functions
1
d
sin−1 (x) = √
dx
1 − x2
d
−1
csc−1 (x) = √
dx
x x2 − 1
d
−1
cos−1 (x) = √
dx
1 − x2
d
1
sec−1 (x) = √
dx
x x2 − 1
d
1
tan−1 (x) =
dx
1 + x2
d
−1
cot−1 (x) =
dx
1 + x2
Example. Find the derivative.
2
(A) y = sin−1 (2x)
(B) y = cos−1 (4x2 )
M151 Notes
©, R.G. Lynch, Texas A&M
(C) y = arctan(sin(4x))
−1 (x)
(D) y = xsec
Section 4.6: Inverse Trigonometric Functions Page 5 of 5