Problem 37 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ \sec ^{2} \theta-2=0 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{\pi}{4} + 2k\pi \) and \( \theta = \frac{3\pi}{4} + 2k\pi \), where \( k \) is an integer.

Step 1: Understand the Trigonometric Identity

The equation is given as \( \sec^2 \theta - 2 = 0 \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \). This means we need to find the values of \( \theta \) for which \( \sec^2 \theta = 2 \).

Step 2: Solve for \( \cos^2 \theta \)

Rewriting the equation \( \sec^2 \theta = 2 \) gives us \( \frac{1}{\cos^2 \theta} = 2 \). By cross-multiplying, we obtain \( \cos^2 \theta = \frac{1}{2} \).

Step 3: Solve for \( \cos \theta \)

Since \( \cos^2 \theta = \frac{1}{2} \), take the square root of both sides to obtain \( \cos \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \).

Step 4: Find General Solutions for \( \theta \)

\( \cos \theta = \frac{\sqrt{2}}{2} \) has solutions \( \theta = \frac{\pi}{4} + 2k\pi \) and \( \cos \theta = -\frac{\sqrt{2}}{2} \) has solutions \( \theta = \frac{3\pi}{4} + 2k\pi \), where \( k \) is an integer. This accounts for all possible angles in the unit circle.

Key Concepts

Understanding the Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). Given the reciprocal nature, the secant function captures the variation where the cosine value approaches zero, leading to larger secant values.

When solving equations involving the secant, such as \( \sec^2 \theta = 2 \), it is crucial to translate the secant in terms of cosine. In this case, the equation equates to \( \frac{1}{\cos^2 \theta} = 2 \), making it easier to solve using familiar methods for the cosine function rather than directly for secant.

Essentials of Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where they are defined. These identities are essential tools for simplifying and solving trigonometric equations.

Some of the most useful identities stem from reciprocal relationships, such as \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). These transformations often aid in solving equations by rewriting terms in a more convenient form.

For example, in solving \( \sec^2 \theta = 2 \), we used the identity to transform \( \sec^2 \theta \) into its equivalent, which then allowed finding \( \cos^2 \theta = \frac{1}{2} \) by simple algebraic manipulations. Understanding and utilizing these identities enable smoother solving of complex trigonometric scenarios.

Exploring the Unit Circle

The unit circle is a crucial concept in trigonometry, representing all angles and their corresponding trigonometric values geometrically. Defined as a circle with a radius of one centered at the origin of a coordinate plane, the unit circle is pivotal for understanding how trigonometric functions behave.

On the unit circle, each angle corresponds to a point whose x-coordinate is \( \cos \theta \) and y-coordinate is \( \sin \theta \). This geometric positioning helps visualize solutions to equations like \( \cos \theta = \pm \frac{\sqrt{2}}{2} \).

For our specific problem, the angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) (and their coterminal angles) correspond to points on the unit circle where the cosine value equals \( \pm \frac{\sqrt{2}}{2} \). This visualization simplifies understanding which angles solve the equation and where they appear on a circle.

Finding General Solutions

When solving trigonometric equations, such as \( \cos \theta = \frac{\sqrt{2}}{2} \) or \( \cos \theta = -\frac{\sqrt{2}}{2} \), finding the general solutions is a critical step. General solutions describe all possible angles satisfying the equation, accounting for the periodic nature of trigonometric functions.

For cosine, which has a period of \( 2\pi \), solutions repeat every \( 2\pi \) radians. Thus, the general solution for \( \cos \theta = \frac{\sqrt{2}}{2} \) is \( \theta = \frac{\pi}{4} + 2k\pi \), where \( k \) is any integer, to include all coterminal angles.

Similarly, for \( \cos \theta = -\frac{\sqrt{2}}{2} \), the angles are \( \theta = \frac{3\pi}{4} + 2k\pi \). These formulas give a complete set of solutions by incorporating the repetitive nature of trigonometric functions identified on the unit circle.