Problem 27 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\sec 4 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{12} + \frac{k\pi}{2} \) and \( x = \frac{5\pi}{12} + \frac{k\pi}{2} \), where \( k \) is any integer.

Step 1: Simplify the Equation

The given equation is \( \sec 4x - 2 = 0 \). To simplify, move 2 to the other side: \( \sec 4x = 2 \).

Step 2: Convert to Cosine Function

Since \( \sec \theta = \frac{1}{\cos \theta} \), we can rewrite \( \sec 4x = 2 \) as \( \frac{1}{\cos 4x} = 2 \). This implies that \( \cos 4x = \frac{1}{2} \).

Step 3: Find General Solutions

The cosine function equals \( \frac{1}{2} \) at angles \( \theta = \frac{\pi}{3} + 2k\pi \) and \( \theta = \frac{5\pi}{3} + 2k\pi \), where \( k \) is an integer.

Step 4: Solve for x

Since \( 4x = \theta \), we have two equations: \( 4x = \frac{\pi}{3} + 2k\pi \) and \( 4x = \frac{5\pi}{3} + 2k\pi \). Solve for \( x \). For the first equation: \( x = \frac{\pi}{12} + \frac{k\pi}{2} \). For the second equation: \( x = \frac{5\pi}{12} + \frac{k\pi}{2} \).

Step 5: Verify Solutions

Using the periodic properties of trigonometric functions, verify that both forms of \( x \) satisfy the original equation. Since cosine has a period of \( 2\pi \), both solutions will satisfy because any integer \( k \) will maintain the equality. Therefore, the solutions are correct.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is an important trigonometric function often used in various mathematical contexts. It is the reciprocal of the cosine function, which means that \( \sec \theta = \frac{1}{\cos \theta} \). This relationship is crucial when solving equations that involve secant, like the one in the original exercise.The secant function is not defined for angles where the cosine function equals zero, since division by zero is undefined. These angles occur at odd multiples of \( \frac{\pi}{2} \), specifically at \( \left(2n+1\right)\frac{\pi}{2} \) where \( n \) is an integer. Therefore, when working with secant, it is essential to consider the domain restrictions.In problems involving the secant function, it is often helpful to first convert the expression into a cosine function format. This simplifies the problem, allowing us to use known values and properties of cosine to find solutions more easily.

Cosine Function

The cosine function, represented as \( \cos \theta \), is one of the fundamental trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the unit circle, it corresponds to the x-coordinate of a point.The cosine function is periodic with a period of \( 2\pi \). This means that it repeats its values in regular intervals, which is an important property when finding solutions to trigonometric equations. In the exercise, recognizing that \( \sec 4x = 2 \) can be rewritten as \( \cos 4x = \frac{1}{2} \) was a key step.Cosine attains the value \( \frac{1}{2} \) at specific standard angles. These angles within one period \( 0 \leq \theta < 2\pi \) are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \). This knowledge is used to find the general solutions for trigonometric equations that involve the cosine function.

General Solutions

Finding general solutions to trigonometric equations means identifying all possible values that satisfy the equation. Because trigonometric functions are periodic, there are generally infinitely many solutions, repeated at regular intervals.For the equation \( \cos 4x = \frac{1}{2} \), the key is to identify the specific angles where this condition holds true. The solutions \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) yield general solutions of the form:

• \( 4x = \frac{\pi}{3} + 2k\pi \)
• \( 4x = \frac{5\pi}{3} + 2k\pi \)

Solving these equations for \( x \) involves dividing through by 4:

• \( x = \frac{\pi}{12} + \frac{k\pi}{2} \)
• \( x = \frac{5\pi}{12} + \frac{k\pi}{2} \)

Here, \( k \) is any integer, allowing us to capture all periodic repetitions of the solutions. This concept of periodicity is critical in trigonometry and helps ensure all solutions are accounted for.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that are true for any value of the variable. They serve as essential tools in simplifying and solving trigonometric equations and proving relationships between different trigonometric expressions.Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal Identities: \( \sec \theta = \frac{1}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \)
• Angle Sum and Difference Identities: for cosine, \( \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \)

In solving the original problem, using the reciprocal identity helped convert the equation \( \sec 4x = 2 \) into a more manageable cosine equation \( \cos 4x = \frac{1}{2} \). By using these identities effectively, solving trigonometric equations becomes a more straightforward task.