/*! 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 82 Find all solutions in radians to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 82

Find all solutions in radians to \(\cos (2 x)=0.89\). Round to the nearest hundredth.

Short Answer

Expert verified
\[x = 0.23 + k\pi\] for integer \( k \)

Step by step solution

01

Isolate the Trigonometric Function

The equation is \(\text{cos}(2x) = 0.89\). The goal is to solve for \x\. First, recall that \2x\ represents the angle for which we need to find the solution.
02

Use the Inverse Cosine Function

To find \2x\, take the inverse cosine (arccos) of 0.89. This will give the principal value of the angle: \[2x = \text{arccos}(0.89)\] Use a calculator to find: \[2x \approx 0.46 \text{ radians}\]
03

Solve for x

Since \2x = 0.46 \text{ radians}\, divide both sides by 2 to solve for \x\: \[x \approx \frac{0.46}{2} \approx 0.23 \text{ radians}\]
04

Consider the Periodicity of the Cosine Function

The cosine function is periodic with a period of \2\pi\. This means that \2x + 2k\pi\ (where \k\ is any integer) will also be solutions. Therefore, \[2x = 0.46 + 2k\text{pi} \] yields additional solutions. For \k = 0\, \[x \approx 0.23 \text{ radians}\] For \k = 1\, \[x \approx 0.23 + \pi \text{ radians} \approx 0.23 + 3.14 \text{ radians} \approx 3.37 \text{ radians}\] For \k = -1\, \[x \approx 0.23 - \pi \text{ radians} \approx 0.23 - 3.14 \text{ radians} \approx -2.91 \text{ radians}\]

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.

cosine function
The cosine function, commonly denoted as \(\text{cos}(x)\), is a fundamental concept in trigonometry. It represents the x-coordinate of a point on the unit circle corresponding to a given angle. This function oscillates between -1 and 1 as the angle changes. A few key properties:
  • Even function: \(\text{cos}(-x) = \text{cos}(x)\)
  • Defined for all real numbers: Any real number input will have a cosine value.
  • Periodicity: This means \(\text{cos}(x + 2\text{蟺}) = \text{cos}(x)\), repeating every \2\text{蟺}\.
Understanding these properties helps solve trigonometric equations like the one given.
inverse trigonometric functions
Inverse trigonometric functions are essential when solving trigonometric equations. For example, arcsine (sin-1), arccosine (cos-1), and arctangent (tan-1) give the angle that corresponds to a given sine, cosine, or tangent value respectively. In our example, we used the arccos function to solve for \2x\ in the equation \( \text{cos}(2x) = 0.89 \). Using a calculator, we find:
  • \text{arccos}(0.89) 鈮� 0.46
    This is the principal value, which is the main solution within the range \[0,蟺\].
  • General solution: To find all possible values, we take into account the periodic nature of the cosine function.
Inverse trigonometric functions help us effectively find these key values.
periodicity in trigonometry
Periodicity refers to the repeating nature of trigonometric functions over intervals. The cosine function has a period of \2\text{蟺}\, meaning its values repeat every \2\text{蟺}\ radians. For our problem:
  • We initially find \2x 鈮� 0.46\ radians.
  • Considering periodicity, we add multiples of \2\text{蟺}\ to find all solutions:
  • \(2x = 0.46 + 2k\text{蟺}\).
  • Solving for \x\: \(x = \frac{0.46 + 2k\text{蟺}}{2}\).
This step extends our solutions beyond the principal value, ensuring we capture all possible angles that satisfy the equation.

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 product and quotient of each pair of complex numbers using trigonometric form. Write your answers in \(a+\) bi form. $$ z_{1}=-3+i, z_{2}=2-i $$

For each rectangular equation, write an equivalent polar equation. $$ x^{2}=4 y $$

For each polar equation, write an equivalent rectangular equation. $$ r=-5 $$

Write \(4-4 i\) in trigonometric form, using degree measure for the argument.

Find the indicated roots. Express answers in the form \(a+b i\) The cube roots of \(\sqrt{3}+i\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure 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.