Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \( \theta = 0 \) for the interval \([0, 2\pi)\).

Step 1: Rewrite the Equation

Given the equation \( \cos \frac{\theta}{2} - 1 = 0 \), rearrange it to form \( \cos \frac{\theta}{2} = 1 \). This tells us that the cosine of half the angle \( \theta \) must equal 1.

Step 2: Solve for the Angle

The cosine function equals 1 at angles like \(0, 2\pi, 4\pi, \ldots\). Therefore, \( \frac{\theta}{2} = 2k\pi \) where \( k \) is an integer.

Step 3: Solve for \( \theta \)

Multiply both sides by 2 to solve for \( \theta \): \( \theta = 4k\pi \).

Step 4: Restrict Solutions to a Given Interval

We need to find solutions for \( \theta \) in the interval \([0, 2\pi)\). Plugging in different integer values for \( k \), we find that only \( k=0 \) results in a value within this interval. For \( k=0 \), \( \theta = 4(0)\pi = 0 \).

Step 5: Verify the Solution

Substitute back into the original equation to verify: \( \cos \frac{0}{2} = \cos 0 = 1 \), which satisfies the equation.

Key Concepts

Cosine Function

The cosine function is an essential part of trigonometry and deals with the ratio of the adjacent side to the hypotenuse in a right-angled triangle. More specifically, it measures how far you move horizontally from an angle along the unit circle, a concept often used in solving trigonometric equations.

For angles like 0, \(2\pi\), or \(4\pi\), the cosine function outputs 1. This is because these angles point directly along the positive x-axis of the unit circle. This property plays a critical role in solving trigonometric equations.

• The cosine of 0 is 1 because it corresponds to the full radius in the positive x-direction.
• This cycle repeats every \(2\pi\), meaning that the cosine function equals 1 at all even multiples of \(\pi\).

Understanding this behavior helps us easily determine the solutions where the cosine function is specifically equal to 1, just like in the equation \( \cos \frac{\theta}{2} = 1 \).

Angle Solutions

Finding angle solutions when dealing with trigonometric equations involves identifying all possible angles that satisfy the given conditions.

In the case of our equation \( \cos \frac{\theta}{2} = 1 \), we look for angles \( \frac{\theta}{2} \) that make the cosine of the angle equal to 1.

Some steps to find angle solutions include:

• Identify the standard angles where the cosine value is 1, such as \(0, 2\pi, 4\pi, \) etc.
• Use these angles to solve for the original variable, in this case, \( \theta \), by manipulating the equation accordingly.
• Account for all possible integer solutions, which can be represented by \(k\) in the formula \( \theta = 4k\pi \).

These steps help pinpoint all the solutions without missing any relevant possibilities that would satisfy the given trigonometric equation.

Interval Notation

Interval notation is a mathematical way to describe a set of numbers between two endpoints. In our problem, the solutions need to be found within the interval \([0, 2\pi)\), a specific range of angle measures.

Key points to remember about interval notation include:

• The brackets \([\ ]\) and \((\ )\) indicate inclusivity and exclusivity, respectively. \([0, 2\pi)\) includes 0 but excludes \(2\pi\).
• It helps in clearly and efficiently expressing solutions within a restricted range.
• Solving for \(\theta\) within this kind of interval involves checking integer values of \(k\) to find which angles fit within \([0, 2\pi)\).

For the given problem, only the angle solution with \(k=0\) fits this interval, making \(\theta = 0\) the valid solution.