Inverse Trig. Functions & Differentiation Section 5.8.
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•Here, you can see that the sine function
y = sin x is not one-to-one.
–Use the Horizontal Line Test.
INVERSE TRIGONOMETRIC FUNCTIONS
•However, here, you can see that
the function f(x) = sin x, ,
is one-to-one.2 2x
INVERSE TRIGONOMETRIC FUNCTIONS
• As the definition of an inverse function states
• that
• we have:
•
• Thus, if -1 ≤ x ≤ 1, sin-1x is the number between and whose sine is x.
1 ( ) ( )f x y f y x
1sin sin and2 2
x y y x y
2 2
INVERSE SINE FUNCTIONS Equation 1
Solve.1
1. arcsin2
y 32. arccos
2
33. sin arctan
4
1sin
2y
6ref
sin QI
6y
3cos
2x
x
6ref
cos QII
5
6x
3arctan
4
x
sin x
3tan
4
3
4
5
oppx
hyp
3
5x
tan QI
QI
• We have:
–This is because , and lies between and .
1 1sin
2 6
sin / 6 1/ 2 / 2 / 2
Example 1 aINVERSE SINE FUNCTIONS
/ 6
• Let , so .
–Then, we can draw a right triangle with angle θ.–So, we deduce from the Pythagorean Theorem
that the third side has length .
1arcsin
3
1sin
3
9 1 2 2
Example 1 bINVERSE SINE FUNCTIONS
–This enables us to read from the triangle that:
INVERSE SINE FUNCTIONS
1 1tan(arcsin ) tan
3 2 2
Example 1b
• In this case, the cancellation equations
for inverse functions become:
1
1
sin (sin ) for2 2
sin(sin ) for 1 1
x x x
x x x
INVERSE SINE FUNCTIONS Equations 2
• The graph is obtained from that of • the restricted sine function by reflection • about the line y = x.
INVERSE SINE FUNCTIONS
•We know that:
–The sine function f is continuous, so the inverse sine function is also continuous.
–The sine function is differentiable, so the inverse sine function is also differentiable (from Section 3.4).
INVERSE SINE FUNCTIONS
•since we know that is sin-1 differentiable, we
can just as easily calculate it by implicit
differentiation as follows.
INVERSE SINE FUNCTIONS
•Let y = sin-1x.
–Then, sin y = x and –π/2 ≤ y ≤ π/2.
–Differentiating sin y = x implicitly with respect to x,we obtain:
INVERSE SINE FUNCTIONS
cos 1
1
cos
dyy
dxdy
dx y
and
• Now, cos y ≥ 0 since –π/2 ≤ y ≤ π/2, so
INVERSE SINE FUNCTIONS
2 2
2
cos 1 sin 1
1 1
cos 1
y y x
dy
dx y x
Therefore
1
2
1(sin ) 1 1
1
dx x
dx x
Formula 3
•Since the domain of the inverse sine
function is [-1, 1], the domain of f is:
INVERSE SINE FUNCTIONS Example 2 a
2 2{ | 1 1 1} { 0 2| }
{ | | 2}
2
|
, 2
x x x x
x x
•Combining Formula 3 with the Chain Rule,
we have:
Example 2 bINVERSE SINE FUNCTIONS
2
2 2
4 2
2 4
1'( ) ( 1)
1 ( 1)
12
1 ( 2 1)
2
2
df x x
dxx
xx x
x
x x
•The inverse cosine function is handled
similarly. –The restricted cosine function f(x) = cos x, 0 ≤ x
≤ π, is one-to-one.
–So, it has an inverse function denoted by cos-1 or arccos.
1cos cos and 0x y y x y
INVERSE COSINE FUNCTIONS Equation 4
• The cancellation equations are:
INVERSE COSINE FUNCTIONS
1cos (cos ) for 0x x x
1cos(cos ) for 1 1x x x
Equation 5
•The inverse cosine function,cos-1,
has domain [-1, 1] and range ,
and is a continuous function.
[0, ]
INVERSE COSINE FUNCTIONS
•Its derivative is given by:
–The formula can be proved by the same method as for Formula 3.
INVERSE COSINE FUNCTIONS Formula 6
1
2
1(cos ) 1 1
1
dx x
dx x
•The inverse tangent function,
tan-1 = arctan, has domain and range
.( / 2, / 2)
INVERSE TANGENT FUNCTIONS
•We know that:
–So, the lines
are vertical asymptotes of the graph of tan.
/ 2x
INVERSE TANGENT FUNCTIONS
( / 2) ( / 2)lim tan lim tanand
x xx x
•The graph of tan-1 is obtained by reflecting
the graph of the restricted tangent function
about the line y = x.
–It follows that the lines y = π/2 and y = -π/2 are horizontal asymptotes of the graph of tan-1.
INVERSE TANGENT FUNCTIONS
Inverse Trig. Functions
None of the 6 basic trig. functions has an inverse unless you restrict their domains.
Function Domain Range
y = arcsin x -1< x < 1 I & IV
y = arccos x -1< x < 1 I & II
y = arctan x < x < I & IV
y= arccot x < x < I & I
y = arcsec x I & II
y = arccsc I & IV
1x
1x
Inverse Properties
f (f –1(x)) = x and f –1(f (x)) = x
Remember that the trig. functions have inverses only in restricted domains.
•
Table 11DERIVATIVES
1 1
2 2
1 1
2 2
1 12 2
1 1(sin ) (csc )
1 1
1 1(cos ) (sec )
1 1
1 1(tan ) (cot )
1 1
d dx x
dx dxx x x
d dx x
dx dxx x x
d dx x
dx x dx x
Derivatives of Inverse Trig. Functions
Let u be a differentiable function of x.
'
2arcsin
1
d uu
dx u
'
2arccos
1
d uu
dx u
2
'
1arctan
u
uu
dx
d
•Each of these formulas can be
combined with the Chain Rule.
•For instance, if u is a differentiable
function of x, then
DERIVATIVES
1
2
12
1(sin )
1and
1(tan )
1
d duu
dx dxu
d duu
dx u dx
Find each derivative with respect to x.
2. arc sec 2f x x
y
2x
1
24 1x
sec 2y xsec tan 2
dyy y
dx
2cos cotdy
y ydx
2
1 12
2 4 1x x
2
1
4 1
dy
dx x x
3. arctany x x y x
arctan x tan x
x
1
21 x
dy dx
dx dx
2sec 1d
dx
21 1d
xdx
2
1
1
d
dx x
2
1arctan
1x x
x
2arctan
1
xx
x
Find each derivative with respect to the given variable.
4. sin arccosh t t
1
t
21 t
sinh t
5. 2arcsin 1f x x
1x 1
21 1x
' cosd
h tdt
cos t sin 1d
dt
21
11
t d
dt
2
1
1
d
dt t
2
1
1 1
t
t
21
t
t
2f x
' 2d
f xdx
sin 1x cos 1
d
dx
1
cos
d
dx
2
2
1 1x
2
1
1 1
d
dx x
Some homework examples:
Write the expression in algebraic form
x3arctansec
Solution: Use the right triangle
xy
yx
3tan
3arctan
Now using the triangle we can find the hyp.
Letthen
y
3x
1
222 9131 xx 291 x
22
911
91sec x
xy
Some homework examples:
Find the derivative of:
2
1
arctanarctan)( xxxf
Let u = 2
1
x
xdx
du
2
1
xxx
xxf
12
1
1
2
1
)( 2
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