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Chapter 0: Problem 52

Write the expression in algebraic form. $$ \sec \left(\sin ^{-1} x\right) $$

Short Answer

Expert verified
The short answer is: \(\sec(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}\).

Step by step solution

01

Define the given expression

We have the expression \(\sec(\sin^{-1}(x))\). Here, \(\sin^{-1}(x)\) denotes the arcsine of x, and \(\sec(\theta)\) is the secant of \(\theta\), which is the reciprocal of the cosine function i.e., \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
02

Let the arcsine of x be 胃

To analyze this expression, we let \(\theta = \sin^{-1}(x)\). By definition, it means that \(\sin(\theta) = x\). Now we have \(\sec(\theta)\) to find.
03

Use the Pythagorean identity

Since we know \(\sin(\theta) = x\), we can use the Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\). We want to find \(\cos(\theta)\), so we will isolate it in the Pythagorean identity: \(\cos^2(\theta) = 1 - \sin^2(\theta)\) Now we substitute \(\sin(\theta) = x\): \(\cos^2(\theta) = 1 - x^2\) Take the square root of both sides to get: \(\cos(\theta) =\pm \sqrt{1-x^2}\) Since 胃 is the arcsine function of x, and the arcsine function has a range of \([- \frac{\pi}{2}, + \frac{\pi}{2}]\), the cosine function will always be positive within this range. This means that we can ignore the negative square root and just use the positive value: \(\cos(\theta) = \sqrt{1-x^2}\)
04

Finding the secant

Recall that the secant is the reciprocal of the cosine function. Thus, to find \(\sec(\theta)\), we will take the reciprocal of \(\cos(\theta) = \sqrt{1-x^2}\). So we have: \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\sqrt{1-x^2}}\) With \(\theta = \sin^{-1}(x)\), the algebraic expression becomes: \(\sec(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}\)

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Most popular questions from this chapter

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ f(x)=2 x+3 ; \quad g(x)=\frac{x-3}{2} $$

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=\sqrt{4-x^{2}}\); shifted horizontally to the right by 2 units, compressed horizontally by a factor of 2, and shifted vertically upward by 1 unit

Find the exact value of the given expression. $$ \sin \left(\sin ^{-1} \frac{1}{\sqrt{2}}\right) $$

a. Describe how you would construct the graph of \(f(|x|)\) from the graph of \(y=f(x)\). b. Use the result of part (a) to sketch the graph of \(y=\sin |x|\).

Write the expression in algebraic form. $$ \cos \left(\sin ^{-1} x\right) $$
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