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Chapter 0: Problem 49

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

Short Answer

Expert verified
The algebraic form of the expression \(\cos(\sin^{-1}(x))\) is \(\sqrt{1 - x^2}\).

Step by step solution

01

Assign the angle

First, we assign the angle \(\theta\) as \(\sin^{-1}(x)\), which means that \(\theta\) is the angle whose sine is \(x\). That is, \[ \sin(\theta) = x \]
02

Use Pythagorean identity

Now, we will use the Pythagorean identity to determine the cosine of \(\theta\): \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] We already know that \(\sin(\theta) = x\), so we can substitute this value into the equation: \[ x^2 + \cos^2(\theta) = 1 \]
03

Solve for \(\cos(\theta)\)

Next, we want to find the algebraic expression for \(\cos(\theta)\). So, we solve for \(\cos(\theta)\) in the Pythagorean identity from Step 2: \[ \cos^2(\theta) = 1 - x^2 \] Then, take the square root of both sides: \[ \cos(\theta) = \pm\sqrt{1 - x^2} \]
04

Determine the appropriate value of \(\cos(\theta)\)

Since we are dealing with the inverse function \(\sin^{-1}(x)\), we know that \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), and within this range of values for \(\theta\), the cosine function is non-negative. Thus, the positive square root is the appropriate value: \[ \cos(\theta) = \sqrt{1 - x^2} \]
05

Substitute \(\theta\) with \(\sin^{-1}(x)\)

Finally, substitute \(\theta = \sin^{-1}(x)\) into the expression for \(\cos(\theta)\) to obtain the algebraic expression for \(\cos(\sin^{-1}(x))\): \[ \cos(\sin^{-1}(x)) = \boxed{\sqrt{1 - x^2}} \]

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Most popular questions from this chapter

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. $$y=x^{2}, \quad y=\left|x^{2}-2 x-1\right|$$ 54\. $$y=\tan x, \quad y=\tan \left(x+\frac{\pi}{3}\ri

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Suppose that \(f\) is a one-to-one function such that \(f(2)=5\). Find \(f^{-1}(5)\).

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