THE INVERSE TRIGONOMETRIC FUNCTIONS.docx
THE INVERSE TRIGONOMETRIC FUNCTIONS
Consider the trigonometric functions and their inverse relationships below:
y = sin(x) and x = sin(y)
y = cos(x) and x = cos(y)
y = tan(x) and x = tan(y)
The inverse relation can be inverse functions if the domain is restricted,
It is conventional to define the inverse functions as follows:
Domain and range [-1, 1] Domain [-1, 1] and range
Domain and range [-1, 1] Domain [-1, 1] and range
| Domain and range (-∞, ∞) Domain (-∞, ∞) and range SummaryFunctionDomainRangey = sin-1(x) = arc sin(x) [-1, 1]y = cos-1(x) = arc cos(x)[-1, 1]y = tan-1(x) = arc tan(x)(-∞, ∞)How to differentiate the inverse trig functions:Given y = sin-1(x) then x = sin (y). = ?Alternatively, we can consider the triangle where sin (y) = x The third side is so if .Likewise given y = cos-1(x) then x = cos (y). ?x = cos (y) OR using the triangle If ____________________________________________________________________________Given y = tan-1(x) then x = tan (y). ?OR If |
SUMMARY
| Function | Derivative |
| y = sin-1(x) | |
| y = cos-1(x) | |
| y = tan-1(x) |
Also
| Function | Derivative |
| y = sin-1f(x) | |
| y = cos-1f(x) | |
| y = tan-1f(x) |
Examples
Find the derivative of each of the following:
(a) y = tan-1(x)
(b) y = tan-1(5x)
(c) y = tan-1(4+x)
OR
(d) y = arcsin(x2 – 3)
In general
If y = arc sin(xn) then
Also
| Function | Derivative |
| y = sin-1f(x) | |
| y = cos-1f(x) | |
| y = tan-1f(x) |
More examples:
(e) y = cos-1(3x-2)
(f) y = tan-1()
(g) y = sin-1(4x2)
(h) y = x2cos-1(x) Using the product rule:
(i) If f(x) = sin-1x + cos-1x, show that f’(x) = 0
Mental work: State the derivatives of
THE INTEGRATION OF TRIGONOMETRIC FUNCTIONS.
Given
| Function | Derivative |
| y = sin-1(x) | |
| y = cos-1(x) | |
| y = tan-1(x) |
It follows that
and
likewise
Examples
(a) Experiment with the substitution u = 2x
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
Sometimes the transformation required before we integrate is a little more complicated.
For example
The rule illustrated here is
Prove this rule:
Examples
1. Show that
(a)
(b)
(c)
2. Use the substitution to show .
3. Show that
(a)
(b)
(c)