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Chapter 8: Problem 11

In Problems 11-26, find the exact value of each expression. \(\sin ^{-1} 0\)

Short Answer

Expert verified
The exact value of \(\text{arc} \sin(0)\) is 0.

Step by step solution

01

Understand the Problem

The goal is to find the exact value of \(\text{arc} \sin(0)\), which is the angle whose sine value is 0.
02

Recall the Range of \(\text{arc} \sin(x)\)

The range of the inverse sine function \(\text{arc} \sin(x)\) is \([-\pi/2, \pi/2]\). This means the angle we are looking for must be within this interval.
03

Use Sine Definition

Recall that \(\text{sin}(\theta) = 0\) when \(\theta = 0\) within the interval \([-\pi/2, \pi/2]\).
04

Conclude the Angle

Since \(\text{sin}(0) = 0\), it follows that \(\text{arc} \sin(0) = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arc Sine

The term 'arc sine,' often written as \(\text{arc} \sin(x)\), represents the inverse sine function.
It's used to find the angle whose sine value equals a given number.
For instance, when we say \(\text{arc} \sin(0)\), we are looking for an angle\( \theta \) such that \(\text{sin}(\theta) = 0\).
Notice that \(\text{arc} \sin(x)\) is only defined for values of x within the interval \([-1, 1]\).
This is because the sine function itself only produces values within this range.
To find the angle corresponding to a sine value, we consider the principal value, which falls within the trigonometric range of the inverse sine function.
Sine Function

The sine function, denoted as \(\text{sin}\), is fundamental in trigonometry.
This function takes an angle as input and outputs the ratio of the opposite side to the hypotenuse in a right-angled triangle.
The general equation is:
\[ \text{sin}(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]
Important properties include periodicity and symmetry.
The sine function is periodic with period \((2\pi)\), meaning \(\text{sin}(\theta + 2\pi k) = \text{sin}(\theta)\) for any integer k.
It is also an odd function where \(\text{sin}(-\theta) = -\text{sin}(\theta)\).
Trigonometric Range

Understanding the range of trigonometric functions, especially the inverse sine or arc sine, is vital in solving related problems.
The range of a function is the set of all possible output values.
For the \(\text{arc} \sin(x)\), the range is \[ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\].
This means any output of the \(\text{arc} \sin(x)\) function will be an angle within \([-\pi/2, \pi/2]\).
In the context of the given problem, to solve \(\text{arc} \sin(0)\), we need to find an angle within \([-\pi/2, \pi/2]\) such that its sine is 0.
Clearly, \(\theta = 0\) satisfies this, making the solution \(\text{arc} \sin(0) = 0\).

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Most popular questions from this chapter

Calculus Show that the difference quotient for \(f(x)=\cos x\) is given by $$ \begin{aligned} \frac{f(x+h)-f(x)}{h} &=\frac{\cos (x+h)-\cos x}{h} \\ &=-\sin x \cdot \frac{\sin h}{h}-\cos x \cdot \frac{1-\cos h}{h} \end{aligned} $$

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