Inverse Trigonometric Derivatives

Definition of the Inverse Trig Functions

Recall that we write 

        arcsin x 

to mean the inverse sin of x restricted to have values between -p/2 and p/2 (Note that sin x does not pass the horizontal line test, hence we need to restrict the domain.)  We define the other five inverse trigonometric functions similarly.

Trig of Arctrig Functions

Example: 

 Find tan(acrsin(x))

The triangle above demonstrates that

        sin(t) = x/1 = opp/hyp.

Hence 

       

  

Since the tangent is 

        opp/adj

We have

Exercise

Simplify

        cos(arctan(2x))          

Derivatives of the Arctrigonometric Functions

Recall that if f and g are inverses, then 

       

What is 

         d
                arctan(x)
        dx

We use the formula:

Since 

                        opp                x
        tan(q) =                 =          
                         adj                1



we have 

           

so that

 

Theorem

Recall that 

        cos x  =  sin(p/2 - x) 

hence 

        arccos x  =  p/2 - arcsin x

so

        d/dx(arccos x) = d/dx[p/2 - arcsin x] 

        = -d/dx[arcsinx] =

Similarly:

        d/dx(arccot(x)) = -1/(1 + x²)

        d/dx(arccscx) 

Example

Find the derivative of 

        cos(arcsinx)

Solution:  

let 

        y = cosu,      u = arcsinx

        y' = -sin u 

        = -sin(arcsin x) = x

We arrive at

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