Problem 24 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$2 \cos x=\sqrt{3}$$

Step-by-Step Solution

Answer

Solutions are \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{11\pi}{6} + 2n\pi \).

Step 1: Isolate the Trigonometric Function

Start by isolating \( \cos x \) on one side of the equation. Divide both sides by 2 to get: \[ \cos x = \frac{\sqrt{3}}{2} \] Now, \( \cos x \) is in its simplest form.

Step 2: Determine Relevant Angles

Recall that \( \cos x = \frac{\sqrt{3}}{2} \) is a well-known cosine value corresponding to certain angles on the unit circle. We know that this value corresponds to angles \( \frac{\pi}{6} \) and \( \frac{11\pi}{6} \) in the range of 0 to \( 2\pi \).

Step 3: Consider General Solutions

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

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, along with sine and tangent. It originates from the relationship between the sides of a right triangle and has applications extending beyond, into calculus and beyond. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. This geometric explanation extends into the unit circle, facilitating our use of cosine in various math problems.

Mathematically represented, it is written as:

• \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

In the context of the unit circle, cosine becomes handy for identifying coordinates. For any angle \( \theta \), the cosine represents the x-coordinate of the point located on the circumference of the unit circle corresponding to this angle. This understanding helps when solving trigonometric equations, such as finding angles that correspond to given cosine values. Cosine values are often found using notable angles, like \( \frac{\pi}{6} \) and \( \frac{11\pi}{6} \), where \( \cos x = \frac{\sqrt{3}}{2} \). This property becomes crucial for discovering solutions on the unit circle.

Unit Circle

The unit circle is a fundamental concept in trigonometry, aiding the understanding of sine, cosine, and tangent functions. It's a circle centered at the origin of a coordinate plane with a radius of one. Because it has a radius of 1, it simplifies the computation of trigonometric ratios, making it an essential tool in solving trigonometric equations.

On the unit circle, angles can be represented in radians or degrees. The entire circumference represents angles from 0 to \( 2\pi \) radians (or 0 to 360 degrees). Each point on this circle is defined by two coordinates that are precisely \( (\cos \theta, \sin \theta) \), where \( \theta \) is the angle formed with the positive x-axis. For example:

• At \( \theta = 0 \), the point is (1,0), so \( \cos 0 = 1 \)
• At \( \theta = \frac{\pi}{6} \), the point is \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \)

This mapping is central for deductions involving trigonometric functions. The values we associate with cosine, sine, or tangent are not just ratios; they can be visualized as coordinates on this circle, making the unit circle a bridge between geometric understanding and analytical computations.

Periodic Functions

Trigonometric functions such as sine and cosine are inherently periodic, meaning they repeat their values in regular intervals. For the cosine function, the interval or period is \( 2\pi \). This periodicity plays a vital role in solving trigonometric equations since it suggests that the solutions recur indefinitely in both directions along the x-axis.

Understanding the periodicity assists in deriving general solutions for trigonometric equations. For instance, when solving \( \cos x = \frac{\sqrt{3}}{2} \), the solutions lie at \( x = \frac{\pi}{6} \) and \( x = \frac{11\pi}{6} \) within the primary cycle of 0 to \( 2\pi \). To account for every solution beyond this interval, the general formula is framed as:

• \( x = \frac{\pi}{6} + 2n\pi \)
• \( x = \frac{11\pi}{6} + 2n\pi \)

where \( n \) is any integer. Each \( n \) embodies a shift by full periods, retaining the cosine value. This model shows the elegant cyclic nature of trigonometric phenomena and why they are so useful in modeling repetitive events in sciences, engineering, and nature.