Sec(sin^-1(x)) Simplify This Expression

[MasonAngus]

Simplify This Expression: Sec(sin^-1(x))

So I understand how to do this problem if instead of X there is some constant from the unit circle, but I get confused when there is just an X. How do I solve this problem? Thanks

 

[MarkFL]

I would look at it this way...\(\displaystyle \theta=\arcsin(x)\) is an angle, such that the sine of this angle is \(\displaystyle \displaystyle x=\frac{x}{1}\). Since the sine of an angle in a right triangle is the opposite over hypotenuse, draw a right triangle, choose one of the smaller angles to represent \(\displaystyle \theta\), then label the opposite side \(\displaystyle x\), the hypotenuse 1, and by Pythagoras we know the adjacent side. Use this then to compute the secant of \(\displaystyle \theta\).

 

[stapel]

Simplify This Expression: Sec(sin^-1(x))

So I understand how to do this problem if instead of X there is some constant from the unit circle, but I get confused when there is just an X. How do I solve this problem? Thanks

Try doing this the same way you did it back when you were taking trigonometry:

Draw the generic right triangle.

Label the angle of interest with some name, such as "A".

Note that you are taking the secant of this angle, so sin^-1(x) = A.

Note that this equation can be restated as sin(A) = x = x/1.

Label the appropriate sides of the right triangle with this information.

Use the Pythagorean Theorem to find the expression for the third side of the triangle.

From the triangle, read off the value of the secant.

Please reply specifying the step at which you are getting stuck. Thank you!

 

[HallsofIvy]

What MarkFL and Stapel have told you is by far the simplest and best way to do this problem.

If you want a more "formal" method, you know, I hope, that "secant" is defined as the reciprocal of the cosine: for any \(\displaystyle \theta\), \(\displaystyle sec(\theta)= \frac{1}{cos(\theta)}\). You should also know that \(\displaystyle cos(\theta)= \sqrt{1- sin^2(\theta)}\). Putting those together, \(\displaystyle sec(\theta)= \frac{1}{\sqrt{1- sin^2(\theta)}}\).

In this problem, "\(\displaystyle \theta\)" is \(\displaystyle sin^{-1}(x)\) so
\(\displaystyle sec(sin^{-1}(x))= \frac{1}{\sqrt{1- sin^2(sin^{-1}(x))}}\).

What is \(\displaystyle sin(sin^{-1}(x))\)?