Problem 36 - Algebra 2 and Trigonometry

Question

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \sec \theta=-1 $$

Step-by-Step Solution

Answer

\(\theta = \pi\) is the smallest angle satisfying \(\sec \theta = -1\).

Step 1: Understanding the Problem

We need to find the angle \( \theta \) such that \( \sec \theta = -1 \). The secant function \( \sec \theta \) is the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \). This means we need \( \cos \theta = -1 \).

Step 2: Finding Angle in Standard Position

The angle \( \theta \) for which \( \cos \theta = -1 \) is known from the unit circle. At \( \theta = \pi \), the cosine value is \(-1\). Therefore, \( \theta = \pi \) is a solution.

Step 3: Checking for Other Solutions

The cos function is periodic with period \( 2\pi \), meaning \( \cos(\theta + 2k\pi) = \cos \theta \) for any integer \( k \). However, since we want the angle with the smallest absolute value, we consider angles \( \theta \) where \(|\theta|\) is minimal.

Step 4: Conclusion: Smallest Absolute Value

Given the periodic nature of the cosine function and the symmetry of the unit circle, the smallest absolute value that satisfies \( \cos \theta = -1 \) is indeed \( \theta = \pi \), as any other equivalent angle will be farther due to addition or subtraction of \( 2\pi \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. This means that:

• \( \sec \theta = \frac{1}{\cos \theta} \)

To explore what this implies, consider the situation where \( \sec \theta = -1 \). This directly suggests that \( \cos \theta = -1 \) since the reciprocal of -1 is -1 itself. When dealing with the secant function, it is important to identify where the corresponding values of the cosine function meet this relationship.Additionally, understanding this reciprocal relationship helps in various scenarios, especially in solving equations and in trigonometric identities. Whenever you solve for \( \sec \theta \) and know the value it represents, you are essentially working with the inverse value of the cosine at that angle.

Cosine Function

The cosine function, \( \cos \theta \), is another key component of trigonometry. It measures the adjacent side over the hypotenuse in a right triangle, or equivalently, can be seen as the horizontal coordinate of a point on the unit circle. For angles on the unit circle:

• \( \cos 0 = 1 \)
• \( \cos \frac{\pi}{2} = 0 \)
• \( \cos \pi = -1 \)
• \( \cos \frac{3\pi}{2} = 0 \)

For the task at hand, we're particularly interested in where \( \cos \theta = -1 \). This value occurs at \( \theta = \pi \) radians. What makes \( \theta = \pi \) significant besides its numerical result, is that it represents directly facing the negative x-axis on the unit circle.Cosine's periodicity, with each full cycle covering \(2\pi\) radians, invites looking at angles beyond \(0\) to \(\pi\) using the property that \( \cos(\theta + 2k\pi) = \cos \theta \) for any integer \(k\). However, for the smallest absolute value requirement, \(\theta = \pi \) is the most direct solution.

Unit Circle

The unit circle is a simple yet powerful concept in trigonometry. It is a circle with a radius of one unit, centered at the origin (0,0) of a coordinate plane. All trigonometric functions can be derived from the unit circle's angles.One of the crucial uses of the unit circle is finding trigonometric function values at given angles. At any point on the unit circle, the x-coordinate represents \( \cos \theta \), while the y-coordinate represents \( \sin \theta \). Hence:

• \( \cos \theta = -1 \) positions you at the point (-1, 0).

When addressing the equation \( \sec \theta = -1 \), this correlates to the angle \( \theta = \pi \) on the unit circle. Understanding this geometrical representation helps in visualizing why certain values occur at specific radian measures.The periodic and symmetrical nature of the unit circle also facilitates recognizing equivalent angle measures, explained by its 360-degree or \( 2\pi \)-radian cycle.

Trigonometric Equations

Trigonometric equations involve finding the angles (usually in radians or degrees) that satisfy specific trigonometric function values. When solving an equation like \( \sec \theta = -1 \), you primarily deal with its equivalent \( \cos \theta = -1 \), stemming from secant's reciprocal definition.Solving trigonometric equations involves:

• Determining the equivalent cosine or sine values when applicable.
• Utilizing references from the unit circle to identify potential angles.
• Considering angles in different periods such that \( \cos(\theta + 2k\pi) = \cos \theta \).

Moreover, achieving the smallest absolute value involves identifying the principal angle first. In this case, as derived, it is \( \theta = \pi \). Understanding these principles can be significant in tackling more complex trigonometric problems and ensuring you find all necessary solutions.