Expert: Steve Holleran - 9/11/2007

Question
Calculus
Inverse functions

Steve, you did a great job in helping me in the past, please help me again.

The question is, find the exact value of the given quantity.

sec [ sin^-1 (the inverse) (-3/4) ]

Here it is again

sec [ sin^-1 (-3/4) ]  

Answer
Hi Jason,

Okay, let's see if I can get through this without thoroughly confusing you.

You're going to want to draw another reference triangle.
You want to diagram sin^-1 (-3/4).  From standard position along the positive x-axis, draw a negative angle into the fourth quadrant.  Then drop the perpendicular back to the x-axis. This angle is the reference angle.

Now, since the sine value is -3/4, and sin = opp/hyp, label the side opposite the reference angle (the altitude to the x-axis) as -3 (due to being in Q4), and the hypotenuse = 4.

Then by Pythagoras, the side along the x-axis is sqrt (16 - 9) = sqrt(7).

Okay, now just read off what you want.  The sec is the reciprocal of the cos, and cos = adj/hyp, so what you want is sec = hyp/adj = 4 / sqrt(7).

Leave it this way, because you were asked for the exact value.

Hope you can understand all this.

Steve