/*! 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 9 Determine all solutions of the g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 9

Determine all solutions of the given equations. Express your answers using radian measure. $$\cos \theta=-1$$

Short Answer

Expert verified
The solutions are \( \theta = \pi + 2n\pi \), where \( n \) is an integer.

Step by step solution

01

Identify where the cosine function equals -1

The cosine function, \( \cos \theta \), is equal to -1 at specific points on the unit circle. Recall that cosine corresponds to the x-coordinate on the unit circle.
02

Determine \( \theta \) for \( \cos \theta = -1 \)

On the unit circle, \( \cos \theta = -1 \) at the angle \( \pi \) radians because it corresponds to the point (-1, 0) on the unit circle.
03

Consider the periodicity of the cosine function

The cosine function is periodic with a period of \( 2\pi \). Therefore, solutions can be expressed as \( \theta = \pi + 2n\pi \), where \( n \) is any integer, to account for all full rotations around the circle.

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 a fundamental concept in trigonometry that links angles and lengths. When we talk about the cosine of an angle, we are referring to the x-coordinate of a point on the unit circle drawn for that angle. The cosine function tells us how far left or right we go along the horizontal axis in relation to a circle. Here's what's important about cosine:
  • It is one of the basic trigonometric functions, alongside sine and tangent.
  • The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
  • In the unit circle, the cosine of an angle gives us the horizontal position of the corresponding point.
Understanding the cosine function is essential for solving various trigonometric equations, such as finding solutions where the function equals -1.
Unit Circle
The unit circle is a crucial tool in trigonometry and helps visualize cosine, sine, and tangent functions. It is called the unit circle because its radius is exactly 1. With its center at the origin of a coordinate plane, it helps clarify the trigonometric functions by providing a graphical representation. Key points about the unit circle:
  • Each angle on the unit circle has a corresponding point, where the x-value is the cosine and the y-value is the sine of that angle.
  • The circle allows us to see angles in both degrees and radians.
  • Important angles like \(0, \pi/2, \pi, 3\pi/2,\text{ and } 2\pi\) are typically emphasized in trigonometry.
  • The point \((-1, 0)\) on the unit circle is where the cosine function equals -1, which occurs at an angle of \(\pi\) radians.
The unit circle helps in comprehending the entire set of solutions for trigonometric equations by visualizing these rotations and angles.
Radian Measure
Radian measure is an alternative to using degrees for measuring angles, often making it easier to work with trigonometric equations. Unlike degrees, which divide a circle into 360 equal parts, radians relate directly to the unit circle:
  • One full rotation around the circle is \(2\pi\) radians. Thus, \(\pi\) radians represents half a rotation, or 180 degrees.
  • Radians offer a more natural calculation for circle-based questions, particularly when dealing with periodic functions.
  • Since the circumference of a unit circle is \(2\pi\), using radian measure harmonizes with this natural properties of circles.
Understanding radian measure allows us to express multiple solutions for trigonometric equations, like expressing angles as \(\theta = \pi + 2n\pi\), where \(n\) is an integer.
Periodicity of Functions
Periodicity refers to the repeating nature of trigonometric functions over specific intervals. The cosine function, like other trigonometric functions, has a set period over which its values repeat.
  • The cosine function has a period of \(2\pi\). This means that after an interval of \(2\pi\) radians, the cosine function returns to its starting value, creating a repeating pattern.
  • This property is crucial because it allows us to account for all possible solutions to a trigonometric equation by adding or subtracting full periods.
  • For instance, when solving for \(\theta\) where \(\cos \theta = -1\), knowing that the solutions are periodic lets us express them as \(\theta = \pi + 2n\pi\). Here, \(\pi\) is where \(\cos \theta = -1\) initially, and \(2n\pi\) represents the repeating nature of the circle through integer \(n\).
Understanding the periodicity of functions is vital to find all potential solutions to problems like these, making it a cornerstone in trigonometric problem-solving.

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

Solve the equations on the interval \([0,2 \pi]\) as follows. Graph the expression on each side of the equation and then zoom in on the intersection points until you are certain of the first three decimal places in each answer. For instance, for Exercise \(53,\) when you graph the two equations \(y=\cos x\)and \(y=0.623\) on the interval \([0,2 \pi],\) you 'll see that there are two intersection points. The \(x\) -coordinates of these points are roots of the equation \(\cos x=0.623\). $$\sin (\cos x)=\sin x$$

Solve the given equations. Consider the equation \(\cos ^{-1} x=\tan ^{-1} x\) (a) Explain why \(x\) cannot be negative or zero. (b) As you can see in the accompanying figure, the graphs of \(y=\cos ^{-1} x\) and \(y=\tan ^{-1} x\) intersect at a point in Quadrant I. By solving the equation \(\cos ^{-1} x=\tan ^{-1} x\) show that the \(x\) -coordinate of this intersection point is given by $$x=\sqrt{\frac{\sqrt{5}-1}{2}}$$ (Graph cant copy) (c) Use the result in part (b) along with your calculator to specify the coordinates of the intersection point.

(a) Use a calculator to verify that the value \(x=\sin \frac{5 \pi}{18}\) is a root of the cubic equation \(8 x^{3}-6 x+1=0\) (b) Use the identity \(\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta\) (from Exercise 40 ) to prove that \(\sin \frac{5 \pi}{18}\) is a root of the equation \(8 x^{3}-6 x+1=0 .\) Hint: In the identity, substitute \(\theta=5 \pi / 18\).

(a) Beginning with the identity \(\cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta\) prove that \(\cos 2 \theta=1-2 \sin ^{2} \theta\) (b) Using the result in part (a), prove that \(\sin ^{2} \theta=(1-\cos 2 \theta) / 2\) (c) Derive the formula for \(\sin (s / 2)\) as follows: using the identity in part (b), replace \(\theta\) with \(s / 2,\) and then take square roots.

In this exercise you will see how certain cubic equations can be solved by using the following identity (which we proved in Example 3 in Section 8.2 ): $$ 4 \cos ^{3} \theta-3 \cos \theta=\cos 3 \theta $$ For example, suppose that we wish to solve the equation $$ 8 x^{3}-6 x-1=0 $$ (2) To transform this equation into a form in which the stated identity is useful, we make the substitution \(x=a \cos \theta,\) where \(a\) is a constant to be determined. With this substitution, equation ( 2 ) can be written $$ 8 a^{3} \cos ^{3} \theta-6 a \cos \theta=1 $$ In equation (3) the coefficient of \(\cos ^{3} \theta\) is \(8 a^{3} .\) since we want this coefficient to be \(4 \text { [as it is in equation }(1)]\), we divide both sides of equation (3) by \(2 a^{3}\) to obtain $$ 4 \cos ^{3} \theta-\frac{3}{a^{2}} \cos \theta=\frac{1}{2 a^{3}} $$ Next, a comparison of equations (4) and (1) leads us to require that \(3 / a^{2}=3 .\) Thus \(a=\pm 1 .\) For convenience we choose \(a=1 ;\) equation (4) then becomes $$ 4 \cos ^{3} \theta-3 \cos \theta=\frac{1}{2} $$ Comparing equation (5) with the identity in (1) leads us to the equation $$ \cos 3 \theta=\frac{1}{2} $$ As you can check, the solutions here are of the form $$ \theta=20^{\circ}+120 k^{\circ} \quad \text { and } \quad \theta=100^{\circ}+120 k^{\circ} $$ Thus \(x=\cos \left(20^{\circ}+120 k^{\circ}\right) \quad\) and \(\quad x=\cos \left(100^{\circ}+120 k^{\circ}\right)\) Now, however, as you can again check, only three of the angles yield distinct values for \(\cos \theta,\) namely, \(\theta=20^{\circ}\) \(\theta=140^{\circ},\) and \(\theta=260^{\circ} .\) Thus the solutions of the equation \(8 x^{3}-6 x-1=0\) are given by \(x=\cos 20^{\circ}, x=\cos 140^{\circ}\) and \(x=\cos 260^{\circ} .\) Note: If you choose \(a=-1,\) your solutions will be equivalent to those we found with \(a=1\) Use the method just described to solve the following equations. (a) \(x^{3}-3 x+1=0\) Answers: \(2 \cos 40^{\circ},-2 \cos 20^{\circ}, 2 \cos 80^{\circ}\) (b) \(x^{3}-36 x-72=0\) (c) \(x^{3}-6 x+4=0 \quad\) Answers: \(2,-1 \pm \sqrt{3}\) (d) \(x^{3}-7 x-7=0\) (Round your answers to three decimal places.)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.