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Chapter 0: Problem 37

Find the exact value of the given expression. $$ \sin ^{-1} \frac{1}{2} $$

Short Answer

Expert verified
The exact value of the given expression is \(\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}\).

Step by step solution

01

Understand the inverse sine function

The inverse sine function, written as \(\sin^{-1}\), is a function that takes a number between -1 and 1 and gives an angle in radians which is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). In other words, it is the function that "undoes" the sine function. So, given a value, it returns the angle in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) whose sine is equal to that value.
02

Identify the value we need to evaluate

In the given expression, we want to find the angle whose sine is \(\frac{1}{2}\): \[ \sin^{-1}\left(\frac{1}{2}\right) \]
03

Determine the angle with sine equal to \(\frac{1}{2}\)

We need to find an angle \(\alpha\) in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that: \[ \sin (\alpha) = \frac{1}{2} \] Using the properties of right triangles, we can find the angle by recalling the sine of specific angles in the unit circle. We know that \(sin(\frac{\pi}{6}) = \frac{1}{2}\). Therefore, the required angle is: \[ \alpha = \frac{\pi}{6} \]
04

Write the final answer

The exact value of the given expression is: \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

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Most popular questions from this chapter

Classify each function as a polynomial function (state its degree), a power function, a rational function, an algebraic function, a trigonometric function, or other. a. \(f(x)=2 x^{3}-3 x^{2}+x-4\) b. \(f(x)=\sqrt[3]{x^{2}}\) c. \(g(x)=\frac{x}{x^{2}-4}\) d. \(f(t)=3 t^{-2}-2 t^{-1}+4\) e. \(h(x)=\frac{\sqrt{x}+1}{\sqrt{x}-1}\) f. \(f(x)=\sin x+\cos x\)

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ \begin{array}{l} f(x)=-2 x^{4}+5 x^{2}-4\\\ \text { 7. } f(x)=\frac{x^{3}}{x^{3}+1} \end{array} $$

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

Determine whether the function is one-to-one. $$ f(x)=\sqrt{1-x} $$

Find \(f^{-1}(a)\) for the function \(f\) and the real number \(a\). $$ f(x)=2 x^{5}+3 x^{3}+2 ; \quad a=2 $$
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