Lecture 6 - University of Houston

[Pages:19]Lecture 6Section 7.7 Inverse Trigonometric Functions Section

7.8 Hyperbolic Sine and Cosine

Jiwen He

1 Inverse Trig Functions

1.1 Inverse Sine

Inverse Since sin-1 x (or arcsin x)

1

domain:[-

1 2

,

1 2

]

range:[-1, 1]

2

sin(sin-1 x) = x 3

4

domain:[-1, 1]

range:[-

1 2

,

1 2

]

Trigonometric Properties

5

sin(sin-1 x) = x tan(sin-1 x) = x

1 - x2 sec(sin-1 x) = 1

1 - x2

Differentiation

cos(sin-1 x) = 1 - x2

cot(sin-1 x) = 1 - x2 x

csc(sin-1 x) = 1 x

Theorem 1.

d sin-1 x = 1 .

dx

1 - x2

Proof. Let y = sin-1 x. Then x = sin y,

d sin-1 x = dx

1

d dy

sin

y

=

1 cos y

=

1 cos(sin-1 x)

=

1

.

1 - x2

Theorem 2.

d sin-1 u = 1

du ,

dx

1 - u2 dx

Integration: u-Substitution

1

du = sin-1 u + C

1 - u2

6

Theorem 3.

g (x) dx = sin-1(g(x)) + C 1 - (g(x))2

Proof Let u = g(x). Then du = g (x) dx,

g (x) dx = 1 du = sin-1 u + C = sin-1(g(x)) + C

1 - (g(x))2

1 - u2

Examples 4.

1

1 dx =

du = sin-1 u+C

= sin-1

x +C.

Note

4 - x2

1 - u2

2

that 4 - x2 = 4

1-

x2 2

.

Let

u

=

x 2

.

Then

du =

1 2

dx.

1

dx =

2x - x2

1 du = sin-1 u + C = sin-1(x - 1) + C. Note that 2x - x2 = 1 - (x2 - 1 - u2

2x + 1) = 1 - (x - 1)2 (complete the square). Let u = x - 1. Then du = dx.

1.2 Inverse Tangent

Inverse Tangent tan-1 x (or arctan x)

7

8

y

=

tan

x

domain:(-

1 2

,

1 2

)

range:(-,

)

 ................
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