/*! 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 8 Find the radian measure of the a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 8

Find the radian measure of the angle with the given degree measure. $$-300^{\circ}$$

Short Answer

Expert verified
The radian measure is \(-\frac{5\pi}{3}\).

Step by step solution

01

Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). This formula arises from the fact that 180 degrees is equivalent to \( \pi \) radians.
02

Substitute the Degree Value

Substitute the degree value, which is \(-300^{\circ}\), into the formula. So, you calculate: \(-300 \times \frac{\pi}{180}\).
03

Simplify the Expression

Simplify the expression \(-300 \times \frac{\pi}{180}\). Cancel the common factors: \(-300\) and \(180\) are both divisible by 60. This simplified form is \(-5 \times \frac{\pi}{3}\).
04

Express the Answer in Radians

After simplification, the radian measure of the angle is \(-\frac{5\pi}{3}\). This is the equivalent radian measure for \(-300^{\circ}\).

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.

Radian Measure
The concept of radian measure is fundamental in mathematics, especially in trigonometry. A radian is a unit of angle measurement that is based on the radius of a circle. Specifically, it鈥檚 the angle created when the arc length is equal to the radius of the circle. In practical terms, this means:
  • A full circle has an angle of 2π radians.
  • An angle that encompasses half of a circle measures π radians.
Radian measure provides a more natural way to relate angles to the geometry of circles. Since radian measure ties directly to the radius and arc of a circle, it is extensively used in calculus and trigonometry.
Unlike degrees, which divide a circle into 360 parts, radians use π to represent half a circle, making many mathematical formulas simpler and more elegant.
Degree to Radian Conversion
Converting degrees to radians is essential when dealing with mathematical functions that use radian measure. The heart of this conversion is the relationship between degrees and radians:
  • 180掳 is equal to π radians.
This means that to convert from degrees to radians, you multiply the degree measurement by \(\frac{\pi}{180}\). If you have an angle in degrees, simply apply the formula:
  • \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\)
For example, converting -300掳 to radians involves multiplying -300 by \(\frac{\pi}{180}\), which allows you to switch from the degree system to the radian system. This conversion process simplifies the integration of geometric and trigonometric problems.
Simplifying Expressions
Simplifying mathematical expressions is crucial for clarifying results and making calculations more manageable. In our example, the task is to simplify the expression \(-300 \times \frac{\pi}{180}\). Here are the key steps:
  • Identify common factors between the numerator and the denominator.
  • Both -300 and 180 can be divided by 60, their greatest common divisor.
This reduction results in the simplified form \(-5 \times \frac{\pi}{3}\) or \(-\frac{5\pi}{3}\).
Simplification not only cleans up the expression but also makes it easier for further mathematical operations. Whether it's performing calculations or drawing insights from an expression, simplification is an invaluable skill.

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 area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ} .\)

To solve a certain problem, you need to find the sine of 4 rad. Your study partner uses his calculator and tells you that $$\sin 4=0.0697564737$$ On your calculator you get $$\sin 4=-0.7568024953$$ What is wrong? What mistake did your partner make?

The Leaning Tower of Pisa The bell tower of the cathedral in Pisa, Italy, leans \(5.6^{\circ}\) from the vertical. A tourist stands \(105 \mathrm{m}\) from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be \(29.2^{\circ} .\) Find the length of the tower to the nearest meter.

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\cos \theta=-\frac{2}{7}, \quad \tan \theta<0$$

Two boats leave the same port at the same time. One travels at a speed of \(30 \mathrm{mi} / \mathrm{h}\) in the direction \(\mathrm{N} 50^{\circ} \mathrm{E}\) and the other travels at a speed of \(26 \mathrm{mi} / \mathrm{h}\) in a direction \(\mathrm{S} 70^{\circ} \mathrm{E}\) (see the figure). How far apart are the two boats after one hour?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.