Problem 10 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) represents a real number. $$ \sqrt{3} \sec x=2 $$

Step-by-Step Solution

Answer

\( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{11\pi}{6} + 2k\pi \), where \( k \) is an integer.

Step 1: Understanding the Equation

We start with the equation \(\sqrt{3} \sec x = 2\). This equation involves the secant function, which is the reciprocal of the cosine function. Therefore, we can rewrite the equation in terms of cosine by recalling that \( \sec x = \frac{1}{\cos x} \).

Step 2: Convert to Cosine

Substitute \( \sec x = \frac{1}{\cos x} \) into the equation: \( \sqrt{3} \cdot \frac{1}{\cos x} = 2\). Simplify it to get \( \sqrt{3} = 2 \cos x \).

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

Divide both sides by 2 to isolate \( \cos x \): \( \cos x = \frac{\sqrt{3}}{2} \).

Step 4: Find General Solutions for \( \cos x = \frac{\sqrt{3}}{2} \)

Cosine of \( x \) equals \( \frac{\sqrt{3}}{2} \) at the angles where \( x = \frac{\pi}{6} + 2k\pi \) or \( x = -\frac{\pi}{6} + 2k\pi \), where \( k \) is an integer. These represent all the angles in standard position that have the required cosine value.

Step 5: Verify and Conclude the Solutions

Verify that substituting \( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{11\pi}{6} + 2k\pi \) (since \( -\frac{\pi}{6} \) corresponds to \( \frac{11\pi}{6} \) in standard angle notation) into the original equation satisfies the equation \( \sqrt{3} \sec x = 2\). Thus, since both configurations satisfy the equation, these are the solutions.

Key Concepts

Secant Function

The secant function, represented as \( \sec x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means that wherever the cosine function has a value of zero, the secant function will not be defined, as division by zero is undefined.

In trigonometry, the secant function is used less frequently than sine, cosine, and tangent, but it is still quite useful in solving various types of equations. Knowing this function's properties helps in problems where particular angle solutions are needed, or when dealing with complex trigonometric identities. For instance, secant positive or negative intervals can guide you to the general behavior of the function.

• Understanding that \( \sec x = \frac{1}{\cos x} \) allows us to rewrite equations involving secant into a form involving cosine, often simplifying the problem.
• Secant and cosine are closely related, and understanding this relationship makes it easier to manage equations involving trigonometric functions.

Cosine Function

The cosine function, denoted as \( \cos x \), is fundamental in trigonometry. It describes the relationship between angles and the sides of a right triangle or the coordinates on the unit circle. The range of the cosine function is from -1 to 1, and it is periodic with a period of \( 2\pi \), meaning the function repeats its values in intervals of \( 2\pi \).

The cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the circle. For cosine values such as \( \frac{\sqrt{3}}{2} \), we often look for standard angles like \( \frac{\pi}{6} \) where cosine holds these special values.

By converting secant to cosine in our equation \( \sqrt{3} \sec x = 2 \), we simplify it to solve for \( \cos x \), which leads directly to our solutions for \( x \). Knowing where cosine achieves particular values helps you solve trigonometric equations efficiently.

• Cosine helps you find angles in trigonometric problems relating to different triangles and circular rotation.
• Recognizing the precise angles for specific cosine values is crucial for life sciences, engineering, and wherever periodic functions are relevant.

General Solutions in Trigonometry

Finding general solutions in trigonometry involves determining all possible angles that satisfy a given trigonometric equation. Since trigonometric functions are periodic, one angle often corresponds to many solutions.

In the given problem, after simplifying the equation to \( \cos x = \frac{\sqrt{3}}{2} \), we identify the standard angles that provide these cosine values. These angles are \( x = \frac{\pi}{6} \) and \( x = -\frac{\pi}{6} \). Due to the periodic nature of cosine, each of these angles repeats every \( 2\pi \) radians.

The general solutions formula incorporates this periodicity by adding multiples of \( 2k\pi \) (where \( k \) is any integer) to these base angles. Therefore:

• For \( \frac{\pi}{6} \), the general solution is \( x = \frac{\pi}{6} + 2k\pi \).
• For \( -\frac{\pi}{6} \), it is often more convenient to express this angle in standard terms as \( x = \frac{11\pi}{6} \), leading to a general solution of \( x = \frac{11\pi}{6} + 2k\pi \).

This approach ensures that we cover all angles corresponding to the desired cosine value across all cycles of the cosine wave. It's a powerful technique in solving cyclic structures or waves and harmonics in physics and engineering.