/*! 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 12 Evaluate the expression without ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 12

Evaluate the expression without using a calculator. $$\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)$$

Short Answer

Expert verified
\(-\frac{3蟺}{4}\) or -135 degrees

Step by step solution

01

Identify quadrant

Since the value to find the arcsine of is negative, it implies that the angle must be in the third or fourth quadrant where sine values are negative.
02

Identify reference angle

We first need to identify an angle in the first or second quadrant that has a sine value of \(\frac{\sqrt{2}}{2}\). This angle is \(\frac{蟺}{4} \) or 45 degrees. Thus, the reference angle for this problem is \(\frac{蟺}{4}\). Remember that the first and second quadrants are the 'mirror' of the third and fourth quadrants, respectively when it comes to sine and cosine values.
03

Find the actual angle

As we established in Step 1, the angle must be in the third or fourth quadrant. Therefore, we subtract the reference angle of \(\frac{蟺}{4}\) from either \(\pi\) (for third quadrant) or \(2\pi\) (for fourth quadrant), to get the corresponding third and fourth quadrant angles: \( \frac{3蟺}{4} \) and \( \frac{7蟺}{4}\) respectively. Since the arcsine function only accepts values in the interval \([-蟺/2 , 蟺/2]\), and \(\frac{-3蟺}{4}\) falls into this interval, our answer is \(\frac{-3蟺}{4}\) or -135 degrees.

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!

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

Area of a Sector of a Circle Find the area of the sector of a circle of radius \(r\) and central angle \(\boldsymbol{\theta}\). $$r=12 \text { millimeters, } \theta=\frac{\pi}{4}$$

Sketch a graph of the function. $$f(x)=\frac{\pi}{2}+\arctan x$$

Converting to \(\mathrm{D}^{\circ} \mathrm{M}^{\prime} \mathrm{S}^{\prime \prime}\) Form \(\quad\) Convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check your answers using a calculator. (a) \(240.6^{\circ}\) (b) \(-145.8^{\circ}\)

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow 1^{-}, \text {the value of } \arcsin x \rightarrow \text{______}$$

Finding the Central Angle Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an are of length \(s\). \(r=14\) feet \(, s=8\) feet
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.