Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \cos ^{2} \gamma+\cos \gamma=0 $$

Step-by-Step Solution

Answer

\([\frac{\pi}{2}, \frac{3\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}]\)

Step 1: Factor the Equation

Start by factoring the given equation: \[ 2\cos^2 \gamma + \cos \gamma = 0 \]We can rewrite this as: \[ \cos \gamma (2\cos \gamma + 1) = 0 \]This equation is factored into two parts: \( \cos \gamma = 0 \) and \( 2\cos \gamma + 1 = 0 \).

Step 2: Solve for \( \cos \gamma = 0 \)

We need to determine where the cosine function equals zero within the interval \([0, 2\pi)\). The cosine of an angle is zero at:\[ \gamma = \frac{\pi}{2}, \frac{3\pi}{2} \]

Step 3: Solve for \( 2\cos \gamma + 1 = 0 \)

Set the inside of the parentheses equal to zero and solve for \( \gamma \):\[ 2\cos \gamma + 1 = 0 \]Subtract 1 from both sides:\[ 2\cos \gamma = -1 \]Divide by 2:\[ \cos \gamma = -\frac{1}{2} \]The cosine of an angle is \(-\frac{1}{2}\) at \[ \gamma = \frac{2\pi}{3}, \frac{4\pi}{3} \]

Step 4: Combine Solutions

Collect all solutions that we have calculated within the given interval \([0, 2\pi)\):\[ \gamma = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3} \]

Key Concepts

Factoring Trigonometric Equations

Factoring trigonometric equations is a useful technique to uncover solutions where trigonometric functions meet specific conditions. Given the equation \( 2 \cos^2 \gamma + \cos \gamma = 0 \), we apply factoring to simplify it. The structure of the equation allows us to take out a common factor, which in this case is \( \cos \gamma \). This simplifies the expression to \( \cos \gamma (2\cos \gamma + 1) = 0 \).

• This factorization is crucial because it splits the equation into two parts: \( \cos \gamma = 0 \) and \( 2\cos \gamma + 1 = 0 \).
• By solving each part separately, we can find the specific values of \( \gamma \) that satisfy the original equation.

Understanding how to factor trigonometric equations can make solving these problems much easier, as it breaks down complex expressions into manageable parts.

Cosine Function

The cosine function is one of the primary trigonometric functions, often abbreviated as \( \cos \). It relates the angle within a right-angled triangle to the ratio of the length of the adjacent side over the hypotenuse. Here, it's crucial to recognize at which angles within the interval \( [0, 2\pi) \) the cosine function possesses specific values.

• For \( \cos \gamma = 0 \), the angles are found at \( \gamma = \frac{\pi}{2} \) and \( \gamma = \frac{3\pi}{2} \).
• These represent the points where the adjacent side is equal to zero, i.e., the terminal side of the angle is vertical.
• Similarly, for \( \cos \gamma = -\frac{1}{2} \), the solutions occur at \( \gamma = \frac{2\pi}{3} \) and \( \gamma = \frac{4\pi}{3} \).

Each value represents a unique angle where the cosine function achieves zero or its half-negative state, crucial for our trigonometric solutions.

Interval Notation

Interval Notation is a concise way to describe a set of numbers, often used with trigonometric equations to define the solutions' range. In our problem, the interval \( [0, 2\pi) \) specifies that solutions must be within this domain.

• \( [0, 2\pi) \) implies that \( 0 \) is included, while \( 2\pi \) is not.
• It’s vital to understand that since trigonometric functions are periodic, solutions might repeat outside this interval.
• However, for this exercise, only consider solutions that lie within it.

Thinking in terms of interval notation will help ensure that your solutions to trigonometric equations are accurately limited to the desired domain.

Trigonometric Solutions

Finding trigonometric solutions involves determining the angles that satisfy a given trigonometric equation within a specified interval. Through factoring and leveraging the properties of the cosine function, we can find solutions to the equation \( 2 \cos^2 \gamma + \cos \gamma = 0 \) over the interval \([0, 2\pi)\).

• From \( \cos \gamma = 0 \), the solutions are \( \gamma = \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).
• From \( 2\cos \gamma + 1 = 0 \), the solutions are \( \gamma = \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \).
• These angles provide all the solutions to the equation within the interval \([0, 2\pi)\).

Identifying these angles is essential for completing problems involving trigonometric equations correctly.