Problem 17 - Algebra and Trigonometry

Question

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ 2 \cos 3 \theta=1 $$

Step-by-Step Solution

Answer

Solutions in \([0, 2\pi)\) are \(\theta = \frac{\pi}{9}, \frac{5\pi}{9}, \frac{7\pi}{9}, \frac{11\pi}{9}, \frac{13\pi}{9}, \frac{17\pi}{9}.\)

Step 1: Solve for Cosine

Start by isolating the cosine term in the equation. Divide both sides of the equation by 2 to get: \[ \cos 3\theta = \frac{1}{2}. \]

Step 2: Identify General Solutions

The cosine function equals \(\frac{1}{2}\) at angles where \(3\theta = \frac{\pi}{3} + 2k\pi\) or \(3\theta = \frac{5\pi}{3} + 2k\pi\), where \(k\) is an integer. These angles are solutions to the equation cos(x) = \(\frac{1}{2}\).

Step 3: Solve for Theta

To find \(\theta\), divide each part of the solutions from Step 2 by 3: \[\theta = \frac{\pi}{9} + \frac{2k\pi}{3},\]\[\theta = \frac{5\pi}{9} + \frac{2k\pi}{3}.\]

Step 4: Identify Solutions within Interval [0, 2π)

Calculate specific solutions that fall within the given range \([0, 2\pi)\) by substituting values for \(k\). Calculate \(\theta\) for smallest positive values ensuring they fall within \([0, 2\pi)\).

Step 5: Compute Specific Values

Taking different values of \(k\), compute: For \(\theta = \frac{\pi}{9} + \frac{2k\pi}{3}\):- \(k = 0\), \(\theta = \frac{\pi}{9}\)- \(k = 1\), \(\theta = \frac{7\pi}{9}\)- \(k = 2\), \(\theta = \frac{13\pi}{9}\)For \(\theta = \frac{5\pi}{9} + \frac{2k\pi}{3}\):- \(k = 0\), \(\theta = \frac{5\pi}{9}\)- \(k = 1\), \(\theta = \frac{11\pi}{9}\)- \(k = 2\), \(\theta = \frac{17\pi}{9}\)

Step 6: Final List of Solutions

The solutions for \(\theta\) in the interval \([0, 2\pi)\) are: \[\theta = \frac{\pi}{9}, \frac{5\pi}{9}, \frac{7\pi}{9}, \frac{11\pi}{9}, \frac{13\pi}{9}, \frac{17\pi}{9}.\]

Key Concepts

Cosine Function

The cosine function is an essential trigonometric function that measures the horizontal projection of an angle in a unit circle. It oscillates between -1 and 1, reflecting its characteristic wave pattern. The function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. Understanding the key characteristics of the cosine function helps in solving equations like \(2 \cos 3\theta = 1\).

When dealing with equations involving cosine, always aim to isolate the cosine term. In our problem, dividing both sides by 2 simplifies the equation to \(\cos 3\theta = \frac{1}{2}\). This value suggests specific angles where the cosine achieves \(\frac{1}{2}\), such as \({\frac{\pi}{3}}\) and \({\frac{5\pi}{3}}\) plus any multiple of \(2\pi\). Remember, trigonometric solutions often involve using identities and periodicity of the sine and cosine functions.

General Solutions

Finding general solutions in trigonometric equations involves identifying all possible angles that satisfy the equation. For \(\cos 3\theta = \frac{1}{2}\), we determine the angles where this condition holds true. This includes angles like \(3\theta = \frac{\pi}{3} + 2k\pi\) and \(3\theta = \frac{5\pi}{3} + 2k\pi\).

Here, \(k\) is an integer, signifying the periodic nature of trigonometric functions. By including \(2k\pi\), we embrace the repeating behavior every \(2\pi\) radians. Solving for \(\theta\), we adjust each general solution by dividing through by 3. Thus, the answers become \({\theta = \frac{\pi}{9} + \frac{2k\pi}{3}}\) and \({\theta = \frac{5\pi}{9} + \frac{2k\pi}{3}}\). General solutions allow us to cover both specific solutions in given ranges and universal solutions across number lines.

Interval Notation

Interval notation is key to representing a range of numbers concisely and clearly. Here, the interval \([0, 2\pi)\) refers to possible solutions for \(\theta\) within one complete circle in radians. The notation \")(\" indicates the boundary is not included (so not precisely \(2\pi\) itself), while \