Problem 33 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ 2 \cos ^{2} \theta-1=0 $$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{4} + 2k\pi\), \(\frac{7\pi}{4} + 2k\pi\), \(\frac{3\pi}{4} + 2k\pi\), \(\frac{5\pi}{4} + 2k\pi\) for integer \(k\).

Step 1: Understand the Equation

The given equation is about trigonometric functions, specifically the cosine function: \(2 \cos^{2} \theta - 1 = 0\). Our goal is to find the values of \(\theta\) that satisfy this equation.

Step 2: Isolate \(\cos^2 \theta\)

First, we add 1 to both sides of the equation to get \(2 \cos^{2} \theta = 1\).

Step 3: Solve for \(\cos^2 \theta\)

Next, we divide both sides of the equation by 2 to isolate \(\cos^2 \theta\), resulting in \(\cos^2 \theta = \frac{1}{2}\).

Step 4: Find \(\cos \theta\)

To find \(\cos \theta\), we take the square root of both sides. This gives us \(\cos \theta = \pm \frac{1}{\sqrt{2}}\). Simplifying, we have \(\cos \theta = \pm \frac{\sqrt{2}}{2}\).

Step 5: Determine \(\theta\)

The general solutions for \(\cos \theta = \frac{\sqrt{2}}{2}\) are \(\theta = \frac{\pi}{4} + 2k\pi\) and \(\theta = \frac{7\pi}{4} + 2k\pi\), where \(k\) is any integer. Similarly, for \(\cos \theta = -\frac{\sqrt{2}}{2}\), the solutions are \(\theta = \frac{3\pi}{4} + 2k\pi\) and \(\theta = \frac{5\pi}{4} + 2k\pi\).

Step 6: Compile the Solutions

The complete set of solutions for the given equation are \(\theta = \frac{\pi}{4} + 2k\pi\), \(\frac{7\pi}{4} + 2k\pi\), \(\frac{3\pi}{4} + 2k\pi\), and \(\frac{5\pi}{4} + 2k\pi\) for integer \(k\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often abbreviated as 'cos'. It is defined on a right-angled triangle as the ratio between the length of the adjacent side to the hypotenuse. In a circle, cosine represents the x-coordinate of a point traced around the unit circle as the angle with the positive x-axis.
Cosine values oscillate between -1 and 1 as the angle changes. Some key properties include:

• The cosine of 0 is 1.
• Cosine is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• The period of the cosine function is \(2\pi\), meaning \( \cos(\theta + 2\pi) = \cos(\theta) \).

Understanding these properties is essential as they allow manipulation of cosine equations and establish key groundwork that aids in solving various mathematical and real-world problems.

General Solutions in Trigonometry

General solutions in trigonometry are solutions that provide all possible answers to a trigonometric equation. Trigonometric functions like sine, cosine, and tangent are periodic, which means they repeat their values at regular intervals. To solve for these equations, it's crucial to account for this periodic nature.
When solving an equation such as \( \cos \theta = a \), where \( a \) is a constant, the general solution form typically looks like:

• \( \theta = 2n\pi + \cos^{-1}(a) \) for \( n \in \mathbb{Z} \).
• This implies that not only are these solutions periodic over full cycles but that they repeat at every integer multiple of \(2\pi\).

Similarly, other forms arise based on the specific trigonometric function and its inherent properties.

Solving Quadratic Trigonometric Equations

Quadratic trigonometric equations involve trigonometric functions raised to powers, similar to quadratics in algebra. The solution process involves methods of algebraic manipulation and utilizing trigonometric identities.
Consider an equation such as \( 2\cos^2 \theta - 1 = 0 \). The first step is to rewrite the equation to isolate \( \cos^2 \theta \), turning it into something manageable like \( \cos^2 \theta = \frac{1}{2} \).

• Next, identify the basic solutions for \( \cos \theta \), here \( \cos \theta = \pm \frac{\sqrt{2}}{2} \).
• To find \( \theta \), use known angle values from the unit circle where cosine takes those values.
• From here, derive the general solution formats, taking the periodic nature of the cosine function into account.

These steps allow one to resolve potentially complex scenarios into manageable solutions, revealing each possible angle that satisfies the original equation fully.