/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Evaluate the following expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 4

Evaluate the following expressions, giving the answer in radians. $$ \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

Short Answer

Expert verified
\(-\frac{\pi}{4}\)

Step by step solution

01

Understanding Inverse Sine Function

The inverse sine function, \( ext{sin}^{-1}(x)\), returns the angle whose sine is \(x\). The result is given within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), known as the principal value range for sine. We need to find the angle whose sine is \(-\frac{\sqrt{2}}{2}\).
02

Identify the Reference Angle

Recall that \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). Therefore, the reference angle for \(-\frac{\sqrt{2}}{2}\) is \(\frac{\pi}{4}\), but negative for the inverse function because we're looking for \(-\frac{\sqrt{2}}{2}\).
03

Determine the Correct Angle in the Principal Range

Since \(x\) in \( ext{sin}^{-1}(x)\) is \(-\frac{\sqrt{2}}{2}\), and we know \( ext{sin}(\theta)\) is \(-\frac{\sqrt{2}}{2}\) at \(-\frac{\pi}{4}\) in the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle we seek must be \(-\frac{\pi}{4}\).
04

Verify the Solution

Check that \( ext{sin}(-\frac{\pi}{4})\) equals \(-\frac{\sqrt{2}}{2}\). Since this is true, \(-\frac{\pi}{4}\) is correct. Thus, \(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Radians
Radians are a way to measure angles based on the radius of a circle. Instead of the usual degrees, which most people are familiar with, radians provide a more natural measure in mathematics.
  • One full circle is equal to \(2\pi\) radians.
  • Half a circle is \(\pi\) radians, equivalent to 180 degrees.
  • A right angle, being a quarter of a circle, measures \(\frac{\pi}{2}\) radians.
Radians are often used with trigonometric functions because they simplify the equations involved.
For example, the sine of an angle can be found using radian measures, especially when dealing with inverse functions like \(\sin^{-1}(x)\). In this context, using radians becomes very handy, especially in calculus or higher-level mathematics. Students often find it easier once they get accustomed to this measure compared to degrees.
Sine Function
The sine function is a fundamental concept in trigonometry. It relates an angle to the length of the opposite side over the hypotenuse in a right triangle.
  • The function is usually written as \(\sin(\theta)\), where \(\theta\) is the angle in radians or degrees.
  • The sine function has a range of values from -1 to 1.

The inverse sine function, denoted as \(\sin^{-1}(x)\), reverses the process, finding the angle for a given sine value.
In this exercise, \(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) finds the angle for which the sine is \(-\frac{\sqrt{2}}{2}\). This specific sine value corresponds to an angle of \(-\frac{\pi}{4}\) in the principal value range for inverse sine, \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Reference Angle
A reference angle is the acute angle a given angle makes with the horizontal axis. These angles play a crucial role in understanding trigonometric functions for their exact values.
  • The angle's sine has the same absolute value as its reference angle but might differ in sign, depending on the quadrant.
  • Reference angles provide a straightforward way to gauge angles outside the standard angle ranges.
In the problem, \(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) requires identifying the angle with \(-\frac{\sqrt{2}}{2}\) as its sine. Knowing that \(\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\), we identify \(\theta\) to be \(-\frac{\pi}{4}\).
This helps us determine \(-\frac{\pi}{4}\) as the needed angle in the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Reference angles simplify finding angles with specific trigonometric values by highlighting the basic relationships.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the first two positive solutions. $$ 3 \cos \left(\frac{\pi}{2} x\right)=-2 $$

Evaluate the following expressions, giving the answer in radians. $$ \tan ^{-1}(-\sqrt{3}) $$

Find a simplified expression for each of the following. $$ \sin \left(\tan ^{-1}(3 x)\right) $$

Find all solutions on the interval \(0 \leq \theta<2 \pi\). $$ 2 \sin (\theta)=-\sqrt{2} $$

Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 64 and 86 degrees during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach 70 degrees?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.