Expert: Steve Holleran - 5/4/2007

Question
Can you help me in solving these three Ttrigonometrical Problems ?

a) Write the equivalent equations in the form of inverse   functions for :
             (i) x= y + cos theta
            (ii) cos y = x^2

b)Give the value of  (i) sin-^1 (cosx) ( consider only an acute angle in your solution )

                    (ii) tan [ sin-^1 (x)]

c) Verify the identity :  

  tan[ 2tan-^1 (x)]= 2tan [ tan-^1 (x)] +[tan -^1 (x^3)]  

Answer
Hello Yuri,

Okay, let's see what can be done here.

A)  For the first one, you want to isolate the term with the trig function. (I'm going to use "A" instead of theta, because its much easier to write.  You can re-substitute at the end):

            x = y + cos A

=>           s - y = cos A    so  A =  arccos(x - y)

                             or  A = cos^-1(x - y)

For the second one, its just one step:

           cos y = x^2   so y = arccos(x^2)


B)  For the first one, lets set this equal to y:

           sin^-1 (cos x) = y

so                 (cos x) = sin y

but we know from the cofunction/complement identities that  sin y = cos (pi/2 - y) so

                  cos x = cos (pi/2 - y)

and then               x = pi/2 - y and y = pi/2 - x.

For the second one, call the expression in the brackets A, since it is basically an angle:

           tan[ sin^-1 x ] = tan [A]

Now if you diagram a first-quadrant angle and label it A, we know that it has a sine of x, which can be thought of as x/1.  Now drop an altitude to the x-axis, creating the reference triangle.  Since sin A = x/1 = opp / hyp,
label the opposite side from A (the altitude) "x" and the hypotenuse "1".  Using pythagorean theorem, the remaining side (along the positive x-axis) = sqrt(1-x^2).

Now just read the tangent value off the triangle:

tan A = opp / adj = x / sqrt(1-x^2).


C)  Something is not correct here.  Please go back and make sure the expression is right, because this is not an identity.  You can check this if you have a graphing calculator:  put the left side in as one function and the right side in as a second function.  In an identity, the graphs should fall on top of one another; here, that does not happen, which leads me to believe that maybe there is a copy mistake.

I hope this was of some help to you.
Let me know what's going on with part C.

Good luck,
Steve Holleran