Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ 2 \cos x=\sqrt{3} $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Trigonometric Function

The given equation is \(2 \cos x = \sqrt{3}\). To isolate \(\cos x\), divide both sides by 2: \[\cos x = \frac{\sqrt{3}}{2} \]

Step 2: Identify Angle(s) with Known Cosine Value

The value \(\cos x = \frac{\sqrt{3}}{2}\) is a known value from trigonometry, corresponding to angles \(x = \frac{\pi}{6}\) and \(x = \frac{11\pi}{6}\), in the interval \([0, 2\pi)\).

Step 3: Consider All Possible General Solutions

The cosine function is periodic with period \(2\pi\). Therefore, the general solutions for \(x\) are given by:\[ x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad x = \frac{11\pi}{6} + 2k\pi \]where \(k\) is any integer.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, representing the relationship between the angle of a right triangle and the ratio of the adjacent side to the hypotenuse. It's one of the basic trigonometric functions, usually denoted as \( \cos \theta \).
Cosine values range from \(-1\) to \(1\), and the function creates a unique output for each angle.

• It helps us understand and solve various problems involving angles and lengths.
• Knowing cosine values for specific angles is crucial, such as \( \frac{\pi}{6} \) and \( \frac{11\pi}{6} \)

For example, \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \), which directly relates to how we solved the given equation. The familiarity with standard cosine values allows one to determine specific angles quickly. We use a unit circle to determine these values, which makes learning cosine especially valuable in comprehending circular and rotational motions.

Periodic Functions

Periodic functions such as the cosine are characterized by repeating their values at regular intervals. This means that for any angle \(x\), the cosine function remains the same if we add a full circle, which mathematically is represented by \(2\pi\) for the cosine.
To better understand:

• The repetition or cycle is called the period.
• For the cosine function, the period is \(2\pi\).

This means that \( \cos(x) = \cos(x + 2\pi) \). Essentially, the same cosine value occurs at infinitely many angles as we can perpetually add or subtract that period. This property is especially useful in solving trigonometric equations because it informs us of all possible solutions across different cycles within the function. The periodic nature is why we can generalize solutions for specific angles over an ongoing range.

General Solutions

When solving trigonometric equations like the one presented, we aim to find all possible angles \(x\) that satisfy the equation. General solutions help us represent these infinite possibilities succinctly. In the context of the cosine function, the periodicity plays a key role.
Here’s how:

• Once we find a specific solution, say \( x = \frac{\pi}{6} \), we add multiples of the period \(2\pi\) to obtain the general solution.
• This principle is expressed as \( x = a + 2k\pi \), where \(a\) is a specific solution and \(k\) is any integer.

By using this concept, we see that the solutions \( x = \frac{\pi}{6} + 2k\pi \) and \(x = \frac{11\pi}{6} + 2k\pi\), encompass all possible \(x\) that meet \( \cos x = \frac{\sqrt{3}}{2} \), accounting for the repeating nature of the cosine function. This concise form helps mathematicians and students solve trigonometric equations efficiently.