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Chapter 8: Problem 29

Solve each equation on the interval \(\cos (2 \theta)=-\frac{1}{2}\)

Short Answer

Expert verified
The solutions are \(60^\circ, 120^\circ, 240^\circ, \ and \ 300^\circ\).

Step by step solution

01

- Understand the equation

First, recognize that the equation \(\cos(2\theta) = -\frac{1}{2}\) involves the cosine function.
02

- Solve for the angle

Identify the values of \(2\theta\) for which \(\cos(2\theta) = -\frac{1}{2} \. The cosine of an angle is -\frac{1}{2} at \ (120^\circ) \, (240^\circ) \, and any angle coterminal with these (i.e., adding multiples of \360^\circ\).
03

- Write down the general solutions

The general solutions for \(2\theta \) are \ 2\theta = 120^\circ + 360^\circ k \ and \ 2\theta = 240^\circ + 360^\circ k \, where \k\ is any integer.
04

- Solve for \(\theta\)

Now, divide each general solution by 2 to solve for \(\theta\). \ \theta = 60^\circ + 180^\circ k \ and \ \theta = 120^\circ + 180^\circ k \.
05

- Identify solutions in the interval

List the solutions that fall within one period of the cosine function, \(0^\circ \leq \theta < 360^\circ\). These solutions are \ 60^\circ, 240^\circ, 120^\circ, \ and \ 300^\circ\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cosine function
The cosine function is fundamental in trigonometry. It relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. Here, the cosine function is expressed as \(\text{cos}(2\theta)\), indicating that the angle involved is \(2\theta\). In our equation, \(\text{cos}(2\theta)=-\frac{1}{2}\), we need to find for which angles \(2\theta\) this relationship holds true. Remember that cosine values repeat in a periodic manner, specifically every \(360^\text{藲}\) or \(2\text{蟺}\) in radians. This periodic nature of cosine helps us find multiple solutions that satisfy the equation.
angles and intervals
When solving trigonometric equations, it's crucial to consider angles and the specific intervals in which they fall. Here, we're tasked with finding solutions within the interval \(0^\text{藲} \text{ to } 360^\text{藲}\). To do this, we first determine where the cosine of an angle equals \(-\frac{1}{2}\). These angles are \(120^\text{藲} \text{ and } 240^\text{藲}\) for one cycle of \(360^\text{藲}\). Because cosine is periodic, any angle coterminal with these, formed by adding multiples of \(360^\text{藲}\), will also satisfy the equation. This periodicity helps us list out the possible angles before narrowing them to our specific interval.
general solutions
The concept of general solutions in trigonometric equations allows us to express all possible angles that solve the equation. Here, for \(\text{cos}(2\theta) = -\frac{1}{2}\), the general solutions are \2\theta = 120^\text{藲} + 360^\text{藲}k\ and \2\theta = 240^\text{藲} + 360^\text{藲}k\, where \(k\) is any integer. By solving for \(\theta\), we divide by 2: \ \theta = 60^\text{藲} + 180^\text{藲}k \ and \ \theta = 120^\text{藲} + 180^\text{藲}k\. This step reveals how angles recur at regular intervals of \(180^\text{藲}\). Finally, to find solutions within our interval, we filter the general solutions: \60^\text{藲}, 120^\text{藲}, 240^\text{藲}, \text{ and } 300^\text{藲}\.

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