Inverse Trigonometric Functions Arc Sine 2多 …y=-1 If xe[-Va,]sinx maps the interval [-Va,Va]onto [-1.1] and is one-to-one.The inverse is called the arc sine function: y=arcsinx,x∈[-l,l] y=sin.e吉， y=arcsin x,xe[-1,1] Figure7.7.2 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Arc Sine If x [− ½π, ½π] , sin x maps the interval [− ½π, ½π] onto [−1, 1] and is one-to-one. The inverse is called the arc sine function: y = arcsin x, x [−1, 1]
Inverse Trigonometric Functions Since the sin and arcsin functions are inverses, (7.7.1) for all x∈[-l,l], sin (arcsinx)=x (7.7.2) for allx[-π，π]， arcsin (sinx)=x. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Since the sin and arcsin functions are inverses
Inverse Trigonometric Functions Example 1 Calculate if defined: 7 (a)arcsin(sin,π) (b)arcsin(sin。π) 16 3 9 b)）sm((arcsin3 (d)arcsin(sn亏） (e)sin (arcsin 2). Main Meny o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Example 1 Calculate if defined: (e) sin (arcsin 2 ). ) 5 9 ) (d) arcsin (sin 3 1 (b) sin (arcsin ) 3 7 ) (b) arcsin (sin 16 1 (a) arcsin (sin
Inverse Trigonometric Functions Example 2 d (arcsin 3x2) dr Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Example 2
Inverse Trigonometric Functions Example 3 Show that for a)0 (7.1.5) dx :arcsin-+C √a2-x Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Example 3 Show that for a 0
Inverse Trigonometric Functions Example 4 dx Evaluate dx Jo V4-x2 Main Meny cn8a
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Example 4 − 3 0 2 . 4 Evaluate dx x dx
Inverse Trigonometric Functions d 1 (7.7.3) (arcsinx)= dx -x The integral counterpart of(7.7.3)reads dx (7.7.4) / =arcsinx+C. dx (7.7.5) √a2-x =arcsin +C Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions The integral counterpart of (7.7.3) reads
Inverse Trigonometric Functions Arc Tangent y=- y=tan-Ix,xreal y=tanx.xe号， Figure7.7.3 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Arc Tangent
Inverse Trigonometric Functions Since the tan and arctan functions are inverses. (7.7.6) for all real numbersx tan (arctanx)=x (7.7.7) for allx∈（-π，π）， arctan(tanx)=x Main Meny o8
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Since the tan and arctan functions are inverses
Inverse Trigonometric Functions Example 5 dx larctan (ax2+bxc) Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Inverse Trigonometric Functions Example 5