Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = 2n\pi ± \frac{\pi}{3}\) or \(x = 2n\pi ± \frac{5\pi}{3}\), where \(n\) is an integer.

Step 1: Identify known cosine values

The value \(\frac{\sqrt{3}}{2}\) corresponds to the cosine of \(60^\circ\) or \(300^\circ\) in degrees, or \(\frac{\pi}{3}\) or \(\frac{5\pi}{3}\) in radians (since the unit circle repeats every \(2\pi\) radians).

Step 2: Formulate all solutions

In general, for \(\cos x = \cos a\), the solutions are \(x = 2n\pi ± a\), where \(n\) is an integer. Here, \(a\) corresponds to \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\). So all solutions \(x\) for this equation will be of those two forms.

Step 3: Write down the final solutions

The solutions are \(x = 2n\pi ± \frac{\pi}{3}\) or \(x = 2n\pi ± \frac{5\pi}{3}\), where \(n\) is an integer.

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry, which helps us understand relationships in triangles and circular paths. It is typically denoted as \( \cos \theta \), where \( \theta \) is an angle.

The cosine function is uniquely periodic and repeats its values every \( 2\pi \) radians. This periodicity means that if \( \cos x = y \), solutions for \( x \) will repeat every \( 2\pi \) interval.

• Cosine values range from -1 to 1.
• The cosine of an angle gives the horizontal coordinate of a point on the unit circle.
• The cosine function is even, meaning \( \cos(-x) = \cos x \).

Understanding the cosine function is crucial when solving trigonometric equations and predicting oscillatory behaviors such as waves and harmonic motions.

Unit Circle

The unit circle is a powerful tool in trigonometry that provides a geometric representation of angles and their sine and cosine values. Essentially, it's a circle with a radius of 1, centered at the origin on the coordinate plane.

Each point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis. As you move around the circle, the x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine.

• A full rotation around the circle is \( 2\pi \) radians.
• Key angles often used include \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
• A help to remember: \(\cos\theta\) reaches its maximum at \(\theta=0\) or any multiple of \(2\pi\).

The unit circle is crucial for converting between degrees and radians and for solving equations like \( \cos x = \frac{\sqrt{3}}{2} \) by identifying corresponding angles.

Radians

Radians are units of angular measure used extensively in trigonometry and calculus. Unlike degrees, which divide a circle into 360 parts, radians make use of the properties of a circle itself.

The radian measure corresponds to the arc length created by an angle on the unit circle. One full rotation (or circle) around the circle is equal to \( 2\pi \) radians.

• \( \pi \) rad is equivalent to 180 degrees.
• Common conversions include: \( \frac{\pi}{3} \) rad equals 60 degrees and \( \frac{5\pi}{3} \) rad equals 300 degrees.
• Using radians simplifies formulas in calculus and analysis.

Understanding radians is crucial for interpreting the periodic nature of trigonometric functions and angles, helping in forming solutions to equations like \( \cos x = \frac{\sqrt{3}}{2} \).

Solving Equations

When faced with trigonometric equations like \( \cos x = \frac{\sqrt{3}}{2} \), a systematic approach is essential for finding all possible solutions. Here’s a breakdown of how to approach it:

First, identify known values by using the unit circle. You recognize that the cosine of angles \( \frac{\pi}{3} \) and \( \frac{5\pi}{3} \) are both \( \frac{\sqrt{3}}{2} \).

• Use the periodic property of cosine, which repeats every \( 2\pi \).
• Calculate general solutions using the formula: \( x = 2n\pi \pm a \), where \( a \) is a specific solution.
• In this case, two forms arise: \( x = 2n\pi \pm \frac{\pi}{3} \) and \( x = 2n\pi \pm \frac{5\pi}{3} \).

Solving trigonometric equations is about understanding the cyclical patterns and using known values to uncover the entire set of solutions effectively.