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Chapter 8: Problem 13

Find the exact value of each expression. \(\sin ^{-1}(-1)\)

Short Answer

Expert verified
-\frac{\pi}{2}

Step by step solution

01

Understand the Problem

The goal is to find the exact value of \(\text{sin}^{-1}(-1)\), which is the angle whose sine is -1.
02

Identify the Sine Function Range

\(\text{sin}^{-1}(x)\) or \(\text{arcsin}(x)\) has a range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means we're looking for an angle \(\theta\) within this interval.
03

Determine the Angle for \(\text{sin}^{-1}(-1)\)

Since we need \(\text{sin}(\theta) = -1\) and \(\theta\) must be in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle that satisfies this condition is \(-\frac{\pi}{2}\).
04

Confirm the Value

Verify that \(\text{sin}(-\frac{\pi}{2}) = -1\). Therefore, \(\text{sin}^{-1}(-1) = -\frac{\pi}{2}\).

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Key Concepts

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

Arcsine Function
The arcsine function, denoted as \(\text{sin}^{-1}(x)\) or \[ \text{arcsin}(x) \], is the inverse of the sine function. This means it helps us find the angle whose sine equals a given value. The range of the arcsine function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). In simple terms, this means that the arcsine function outputs angles strictly within this interval.
When working with the arcsine function, it's important to remember that:
  • \(\text{arcsin}(x)\) returns an angle \(\theta\).
  • The sine of this angle is equal to \(\text{arcsin}(x)\).
  • The angle \(\theta\) is always between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
This understanding is crucial for solving problems where you need to determine angles based on sine values.
Sine Values
Understanding sine values is fundamental. The sine function gives the y-coordinate of an angle on the unit circle. For any angle \(\theta\), \[ \text{sin}(\theta) \] ranges between -1 and 1.
Let's recap some key sine values for commonly recognized angles:
  • \(\text{sin}(0) = 0\)
  • \(\text{sin}(\frac{\pi}{6}) = \frac{1}{2}\)
  • \(\text{sin}(\frac{\pi}{2}) = 1\)
  • \(\text{sin}(-\frac{\pi}{2}) = -1\)
In the context of our problem, \(\text{sin}^{-1}(-1)\), it is known that the sine of -1 occurs at the angle \(-\frac{\pi}{2}\), which is within the range of the arcsine function.
Unit Circle
The unit circle is a powerful tool in trigonometry that helps visualize the sine, cosine, and other trigonometric functions. It is a circle with a radius of 1 centered at the origin of the coordinate plane.
Let's explore how the unit circle helps us understand the sine function and its inverse:
  • Each point on the unit circle corresponds to an angle \(\theta\) measured from the positive x-axis.
  • The sine of an angle \(\theta\) is the y-coordinate of its corresponding point on the unit circle.
  • For example, the point corresponding to \(\theta = \frac{\pi}{2}\) has coordinates (0, 1), so \(\text{sin}(\frac{\pi}{2}) = 1\).
The unit circle not only illustrates the values of trigonometric functions but also aids in understanding their ranges and the specific intervals where their inverses operate. In our problem, we utilize this circle to see that the angle whose sine value is -1 is indeed \(-\frac{\pi}{2}\), fitting perfectly within the arcsine function's range.

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Most popular questions from this chapter

Find the average rate of change of \(f(x)=\log _{2} x\) from 4 to 16 .

Use the following discussion. The formula $$ D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right] $$ Approximate the number of hours of daylight in New York, New York \(\left(40^{\circ} 45^{\prime}\right.\) north latitude \()\), for the following dates: (a) Summer solstice \(\left(i=23.5^{\circ}\right)\) (b) Vernal equinox \(\left(i=0^{\circ}\right)\) (c) July \(4\left(i=22^{\circ} 48^{\prime}\right)\)

Establish each identity. $$ \frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}=\frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta} $$

Write each trigonometric expression as an algebraic expression containing u and \(v .\) Give the restrictions required on \(u\) and \(v\). $$ \cos \left(\cos ^{-1} u+\sin ^{-1} v\right) $$

Establish each identity. $$ \cos (\alpha-\beta) \cos (\alpha+\beta)=\cos ^{2} \alpha-\sin ^{2} \beta $$
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