/*! 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 74 In Exercises 73-78, solve the tr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 74

In Exercises 73-78, solve the trigonometric equation. $$ 6 \cos x-2=1 $$

Short Answer

Expert verified
The solutions to the equation are \(x = \pi/3\) and \(x = 5\pi/3\).

Step by step solution

01

Isolate the Trigonometric Function

To isolate the cosine function, we first have to get rid of any constants associated with it. In this case, we can add 2 to both sides of the equation which converted the equation to \(6 \cos x = 3\). Then, we'll divide both sides by 6 to get \(\cos x = 3/6 = 1/2\).
02

Use the Inverse Cosine Function

Now that we've isolated the cosine function, we can find \(x\) by using the inverse cosine function. We apply it to both sides of the equation, getting \(x = \cos^{-1}(1/2)\).
03

Find the Value of x

Now we can find the value of \(x\) using the inverse cosine function. As per unit circle trigonometry, \(\cos^{-1}(1/2)\) equals to \(\pi/3\) and \(5\pi/3\). So, \(x = \pi/3\) and \(x = 5\pi/3\).

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 is one of the fundamental functions in trigonometry, symbolized as \( \cos \). It relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. In trigonometry, the cosine of an angle \( x \) can also be understood through the unit circle. The cosine function is periodic with a period of \( 2\pi \), meaning that it repeats its values every \( 2\pi \) units of rotation.
  • The cosine function presents values ranging from -1 to 1.
  • A key property is that \( \cos(-x) = \cos(x) \), showcasing its even function traits.
  • It is often used to determine how much of a rotation lies along the horizontal axis in circular motion.
Understanding the cosine function can help solve equations by isolating it and then using other trigonometric properties related to inverse functions.
Unit Circle
The unit circle is a crucial concept in trigonometry, providing a geometric representation of trigonometric functions like sine and cosine. It is a circle of radius 1 centered at the origin of a coordinate plane. The position on the unit circle correlates to angles and allows for a visual understanding of sine, cosine, and tangent.
  • In the unit circle, the x-coordinate of a point corresponds to the cosine of the angle.
  • The y-coordinate corresponds to the sine of the angle.
  • The circle helps identify angle measurements in both radians and degrees.
  • Key angles to remember in radians for cosine are \( 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \).
The unit circle simplifies solving trigonometric equations by providing one of the fundamental resources for determining these key values.
Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find angles when given specific trigonometric values. For cosine, the inverse is denoted as \( \cos^{-1} \) or sometimes as \( \arccos \). These functions effectively reverse what the traditional trigonometric functions do, converting from function value back to angle.
  • Inverse cosine helps find the angle whose cosine is a specified number.
  • Its range is typically limited to the principal range of \([0, \pi]\) to ensure the function's output is well-defined.
  • When you see \( x = \cos^{-1}(1/2) \), it implies finding angles like \( \pi/3 \) and \( 5\pi/3 \).
Grasping inverse trigonometric functions is essential to solving trigonometric equations, providing a powerful tool to find angles and relationships in various contexts.

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

\(r^{2}=9 \sin 2 \theta\)

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi\). Show that the equation of the rotated graph is \(r=f(\theta-\phi) .\)

\(r=8 \sin \theta \cos ^{2} \theta\)

\(r=\cos 2 \theta\)

\(r^{2}=36 \sin 2 \theta\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.