/*! 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 69 $$ \text { Find } \sin ^{-1}(-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 69

$$ \text { Find } \sin ^{-1}(-1 / 2) \text { in degrees. } $$

Short Answer

Expert verified
The solution is \(-30^\text{o}\).

Step by step solution

01

Understand the Problem

The task is to find the angle in degrees whose sine value is \(-\frac{1}{2}\). This angle will lie within the range of the inverse sine function, which is \(-90^\text{o} \text{ to } 90^\text{o}\).
02

Recall the Reference Angle

Recall that the sine of \(30^\text{o}\) is \(-\frac{1}{2}\), but since we are dealing with \(-\frac{1}{2}\), we need the angle in the fourth quadrant. In the unit circle, \( \theta = -30^\text{o}\).
03

Convert to Positive Angle

Since \(-30^\text{o}\) can also be expressed positively within the range of \(-90^\text{o} \text{ to } 90^\text{o}\), the final angle remains \(-30^\text{o}\).

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.

inverse sine function
The inverse sine function is denoted as \(\text{sin}^{-1}(x)\). It is also known as arcsine and returns the angle whose sine is a given number. The function has a range of [-90掳 to 90掳] or \([-\frac{\pi}{2}, \frac{\pi}{2}]\) in radians. This is because sine function values only once return to original values within this interval.
unit circle
To understand angles and trigonometric functions like sine, it's important to know the unit circle. The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. An angle in the unit circle is measured from the positive x-axis. The sine function corresponds to the y-coordinate of the point where the angle's terminal side intersects the circle. For example, at 30掳, \(\text{sin}(30^\text{o}) = \frac{1}{2}\). When dealing with negative values, the inverse sine function will return an angle in the fourth quadrant (if negative) since sine is negative in the third and fourth quadrants.
angle conversion
Converting angles between radians and degrees is very important in trigonometry. The two units are related by the formula: 180掳 = \(\pi\) radians. For example, to convert -30掳 to radians, \(-30掳 \times \frac{\pi}{180^\text{o}} = -\frac{\pi}{6}\). Understanding conversions help, especially when using inverse trigonometric functions, which might require solutions in either degrees or radians.

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

True or false? Do not use a calculator. $$ \sin \left(150^{\circ}\right)=\sin \left(30^{\circ}\right) $$

True or false? Do not use a calculator. $$ \sin \left(91^{\circ}\right)=-\sin \left(89^{\circ}\right) $$

Accurately draw a small \(45-45-90\) triangle on paper or cardboard and cut it out. Label the angles \(45^{\circ}, 45^{\circ},\) and \(90^{\circ} .\) Label the sides \(1,1,\) and \(\sqrt{2}\). Do the same for a \(30-60-90\) triangle. Label its angles \(30^{\circ}, 60^{\circ}\) and \(90^{\circ} .\) Label its sides \(1,2,\) and \(\sqrt{3}\).

Use a calculator to find the value of each function. Round answers to four decimal places. $$ \csc \left(49^{\circ} 13^{\prime}\right) $$

Solve each problem. Spacing Between Teeth The length of an arc intercepted by a central angle of \(\theta\) radians in a circle of radius \(r\) is \(r \theta .\) The length of the chord, \(c\), joining the endpoints of that arc is given by \(c=r \sqrt{2-2 \cos \theta}\). Find the actual distance between the tips of two adjacent teeth on a 12 -in.-diameter carbide-tipped circular saw blade with 22 equally spaced teeth. Compare your answer with the length of a circular arc joining two adjacent teeth on a circle 12 in. in diameter. Round to the nearest thou- sandth.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

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.