Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ 2 \cos 2 \theta-\sqrt{3}=0 $$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{12} + n\pi\) and \(\frac{11\pi}{12} + n\pi\) for integers \(n\).

Step 1: Isolate the Trigonometric Function

We start with the equation \(2 \cos 2\theta - \sqrt{3} = 0\). To isolate \(\cos 2\theta\), add \(\sqrt{3}\) to both sides, giving us \(2 \cos 2\theta = \sqrt{3}\).

Step 2: Solve for \(\cos 2\theta\)

Divide both sides of the equation by 2: \(\cos 2\theta = \frac{\sqrt{3}}{2}\).

Step 3: Find General Solutions

The cosine function \(\cos x = \frac{\sqrt{3}}{2}\) has solutions at specific angles. These angles are \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{11\pi}{6} + 2n\pi\) for any integer \(n\), because these correspond to the cosine values of \(\frac{\sqrt{3}}{2}\) on the unit circle.

Step 4: Relate Solutions Back to \(2\theta\)

Since we have \(x = 2\theta\), substitute to get \(2\theta = \frac{\pi}{6} + 2n\pi\) and \(2\theta = \frac{11\pi}{6} + 2n\pi\).

Step 5: Solve for \(\theta\)

Divide each equation by 2 to find \(\theta\): - \(\theta = \frac{\pi}{12} + n\pi\) - \(\theta = \frac{11\pi}{12} + n\pi\) for integers \(n\). These are the general solutions for \(\theta\).

Key Concepts

Cosine Function

The cosine function is one of the most fundamental trigonometric functions. It is commonly written as \( \cos(\theta) \) and it provides the horizontal coordinate of a point on the unit circle.

• The cosine of an angle shows how much of the angle extends in the horizontal direction.
• Its values range from -1 to 1.

Understanding the behavior of the cosine function is crucial for solving trigonometric equations.The cosine of a double angle, such as \(2\theta\), follows a specific pattern. This pattern can help simplify and solve equations by reducing the double angle to a more manageable form. For our exercise, the equation becomes simpler when recognizing that \(\cos 2\theta = \frac{\sqrt{3}}{2}\), providing a foundation to solve for \(\theta\).

Unit Circle

The unit circle is a powerful tool in trigonometry that helps visualize angles and their corresponding trigonometric values. It is a circle with a radius of 1, centered at the origin of a coordinate plane.

• Angles are measured in radians from the positive x-axis around the circle.
• It provides the exact values of trigonometric functions at key angles.

On the unit circle, common angles like \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\) give specific cosine values, which simplifies the problem of finding solutions to trigonometric equations.The problem \(2\theta\) maps the angle \(\theta\) around the circle twice, doubling the possible solutions within any cycle. This allows us to work out solutions using the points where the cosine value equals \(\frac{\sqrt{3}}{2}\).

General Solutions

Finding the general solutions of a trigonometric equation involves identifying all possible angles that satisfy the equation.

• Trigonometric functions like cosine are periodic, meaning they repeat their values in regular intervals.
• For cosine, the period is \(2\pi\), which means every \(2\pi\) radians, the function starts over.

In our exercise, we find the general solutions for \(2\theta\) by identifying both primary and secondary angles that satisfy the equation. The primary angle is \(\frac{\pi}{6}\), and another equivalent angle in the unit circle’s cycle is \(\frac{11\pi}{6}\). Each of these angles can generate an infinite number of solutions by adding integer multiples of the period, resulting in the formulas \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{11\pi}{6} + 2n\pi\), where \(n\) is any integer.

Angle Solutions

To find the specific angle solutions for \(\theta\), it's necessary to go through each of the general solutions found for \(2\theta\) and adjust them accordingly.

• Since \(x = 2\theta\), we need to divide each solution by 2 to solve for \(\theta\).
• This results in resolving both for the primary and secondary angles.

For our particular equation, starting from \(2\theta = \frac{\pi}{6} + 2n\pi\) and \(2\theta = \frac{11\pi}{6} + 2n\pi\), dividing by 2 gives the actual angle solutions: \(\theta = \frac{\pi}{12} + n\pi\) and \(\theta = \frac{11\pi}{12} + n\pi\). These solutions account for the cyclical nature of angle rotations on the unit circle while providing all possible solutions for \(\theta\).