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Chapter 4: Problem 6

Evaluate the expression without using a calculator. $$ \arcsin 0 $$

Short Answer

Expert verified
\(\arcsin 0 = 0\)

Step by step solution

01

Understanding \(\arcsin\)

The arcsine function, denoted by \(\arcsin(x)\) or \(\sin^{-1}(x)\), is the inverse of the sine function. It returns the angle whose sine is a given number.
02

Remembering the definition of sine

The definition of sine in a unit circle is the y-coordinate. Therefore, the sine of an angle \(胃\) is 0 when the y-coordinate in the unit circle is 0.
03

Evaluating \(\arcsin 0\)

The y-coordinate is 0 at two points in the unit circle: when \(胃 = 0\) and when \(胃=\pi\). However, by definition, \(\arcsin x\) returns values in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, \(\arcsin 0 = 0\).

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Most popular questions from this chapter

Evaluate the expression without using a calculator. $$ \arctan (1) $$

Graph the functions \(f\) and \(g\). Use the graphs to make a conjecture about the relationship between the functions. $$ f(x)=\sin x+\cos \left(x+\frac{\pi}{2}\right), \quad g(x)=0 $$

Use a graphing utility to graph the function. Include two full periods. $$ y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right) $$

Sketch the graph of the function. Include two full periods. $$ y=2 \csc (x-\pi) $$

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin (-0.75) $$
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