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Chapter 4: Problem 53

In Exercises 49-56, use a graph to solve the equation on the interval \([-2\pi, 2\pi]\). \(sec\ x =\ -2\)

Short Answer

Expert verified
The solutions for \(sec\ x =\ -2\) in the interval [-2蟺, 2蟺] are \(x = -2\pi/3, -4\pi/3, 2\pi/3, 4\pi/3\).

Step by step solution

01

Understanding the secant function

Recall that \(sec x = 1/cos x\). The secant function is undefined wherever cos x is equal to zero, i.e., at \( -(2n+1)\pi/2 \) and \( (2n+1)\pi/2 \) for any integer n. Also, secant values range from \( -\infty \) to \( -1 \) and from 1 to \( \infty \).
02

Solving \(sec x = -2\)

Considering the given equation \(sec x = -2\), we need to solve this in terms of cosine as \(cos x = -1/2\). From the unit circle or the graph of cosine function, it is known that cosine values are -1/2 at \(2\pi/3\) and \(4\pi/3\) in the interval [0, 2蟺]. However, since we are considering the interval [-2蟺, 2蟺], we should consider the angles in the fourth quadrant of the negative side too, which are \(-2\pi/3\) and \(-4\pi/3\).
03

Finalizing the solution

While considering the interval [-2蟺, 2蟺], the solution to \(sec x = -2\) or \(cos x = -1/2\) are \(x = -2\pi/3, -4\pi/3, 2\pi/3, 4\pi/3\).

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Key Concepts

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

Secant Function
The secant function, denoted as \( \sec x \), is closely related to cosine. In fact, the secant function is the reciprocal of the cosine function. This means that \( \sec x = \frac{1}{\cos x} \). Because the secant is dependent on the value of cosine, it becomes undefined where cosine equals zero.
This occurs at intervals defined by \(-(2n+1)\pi/2\) and an equivalent positive interval \((2n+1)\pi/2\), where \( n \) is an integer. Think of these points as vertical asymptotes on the graph of the secant function. Here's what you need to know:
  • The secant function is undefined whenever cosine is zero.
  • Secant values only exist between \( -\infty \) to \(-1 \) and from \(1\) to \( \infty \).
  • These characteristics make secant an important function in trigonometry, especially when solving equations.
Cosine
Cosine is a fundamental trigonometric function often abbreviated as 'cos'. It is a periodic function, meaning it repeats its values in regular intervals.
The cosine function is intrinsically linked to the geometry of a circle, which is why it is often visualized using the unit circle.
Some key properties of cosine include:
  • For any angle \(x\), \( \cos x \) is the x-coordinate of the corresponding point on the unit circle.
  • Cosine function range is between -1 and 1.
  • Common angles where cosine has specific values such as \( \cos(\pi) = -1 \) and \( \cos(0) = 1 \) often form the basis of solving trigonometric equations.
The problem given is rooted in transforming a secant equation to a cosine equation, as seen by the relation \( \sec x = -2 \rightarrow \cos x = -\frac{1}{2} \). This transformation highlights the interconnectedness of secant and cosine.
Unit Circle
The unit circle is a powerful tool in trigonometry, representing angles and trigonometric functions on a coordinate plane. Every point on the unit circle corresponds to the coordinates \((\cos x, \sin x)\), where \(x\) is the angle from the positive x-axis.
This means:
  • The x-coordinate directly gives the cosine of that angle.
  • Angles are typically measured in radians, and the full circle is \(2\pi\) radians.
  • Key angles and their cosine values, especially 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) radians, help to quickly solve equations.
The exercise involves finding when cosine equals -\(\frac{1}{2}\). The unit circle helps us identify these specific angles. On the unit circle, cosine equals -\(\frac{1}{2}\) at \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) in the first rotation (0 to \(2\pi\)), with equivalent negative angles at \(-\frac{2\pi}{3}\) and \(-\frac{4\pi}{3}\) in the interval \([-2\pi, 0]\).
Interval Solutions
Interval solutions refer to finding solutions of an equation within a specified range. For trigonometric equations like \(sec x = -2\), it鈥檚 crucial to determine solutions within the interval \([-2\pi, 2\pi]\).
This involves several steps:
  • Convert the given secant equation to a cosine equation: \(sec x = -2 \rightarrow cos x = -\frac{1}{2}\)
  • Determine where \(cos x = -\frac{1}{2}\) occurs within the interval by using tools like the unit circle.
  • Identify specific angle solutions. In this problem: \(x = -2\pi/3, -4\pi/3, 2\pi/3, 4\pi/3\).
By considering the periodic nature of cosine, additional solutions can often be found by adding or subtracting full rotations (\(2\pi\)) when within the interval bounds. This systematic approach ensures all solutions are covered, providing a comprehensive understanding of the trigonometric equation.

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