Problem 5 - Precalculus : Building Concepts and Connections

Question

Find the zero(s) of \(f(\theta)=\cos 2 \theta\) in the interval \([0, \pi]\)

Step-by-Step Solution

Answer

The zeroes of the function \(f(\theta)=\cos 2 \theta\) in the interval \([0, \pi]\) are \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\).

Step 1: Set the Function Equal to Zero

To find the zero(s) of the function, start by setting the function equal to zero: \(\cos 2 \theta = 0\).

Step 2: Solve for \(\theta\)

The function \(\cos \theta\) equals zero at \(\theta=\frac{\pi}{2} + n\pi\) where \(n\) is an integer. The argument of the cosine function in this problem is \(2\theta\) instead of \(\theta\). Thus, set \(2\theta=\frac{\pi}{2} + n\pi\). Solving for \(\theta\) gives \(\theta = \frac{\pi}{4} + \frac{n\pi}{2}\). Now, it's necessary to determine which of those solutions are within the interval \([0, \pi]\).

Step 3: Find Valid Solutions

On plugging in values of \(n\) we get: For \(n=0\), \(\theta = \frac{\pi}{4}\), which is part of the desired interval. For \(n=1\), \(\theta = \frac{3\pi}{4}\), which is also part of the desired interval. For \(n=2\), \(\theta = \frac{5\pi}{4}\), which is not part of the desired interval. Therefore, the zeroes of the function in the interval \([0,\pi]\) are \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\).

Key Concepts

Trigonometric Zeros Interval

Understanding the trigonometric zeros interval is crucial when working with trigonometric functions, as these are the points where the function value is zero. For any trigonometric function, such as sine or cosine, there are specific intervals where the function's graph crosses the x-axis, indicating a zero value.

In the context of our exercise, the zero(s) of the function \(f(\theta)=\cos 2\theta\) within the interval \([0, \pi]\) were found by setting the function equal to zero and solving for \(\theta\). It's always important to account for the periodic nature of trigonometric functions when solving for zeros. Since trigonometric functions are cyclical, they have an infinite number of zeros at regular intervals, usually expressed with the addition of \(n\pi\), where \(n\) is an integer that represents the number of cycles.

Cosine Function Zeros

The cosine function has a characteristic wave pattern that repeats every \(2\pi\) radians or 360 degrees. For the cosine function, the zeros occur where the angle corresponds to an odd multiple of \(\frac{\pi}{2}\). This can be represented as the set of angles where \(\cos \theta=0\) with \(\theta=\frac{\pi}{2} + n\pi\), where \(n\) is any integer.

In our exercise, the zero(s) for \(f(\theta)=\cos 2\theta\) are found under this principle but with adjustments due to the argument being \(2\theta\). Remember, the zeros of the cosine function provide a foundational concept for solving various trigonometric equations and also for understanding the behavior of waves and oscillations in fields like physics and engineering.

Solving Trigonometric Equations

When solving trigonometric equations, one must be familiar with the basic identities and values of trigonometric functions. The strategy often involves isolating the trigonometric function and then expressing the solutions in terms of \(\pi\) by considering the function's period.

With the given exercise, solving the equation \(\cos 2\theta = 0\) demanded the knowledge of where cosine equals zero and then algebraic manipulation to isolate \(\theta\). After finding the general solution, limiting the results to the specified interval, which was \([0, \pi]\) in this case, produced the specific solutions we were seeking. Moreover, recognizing that only certain values of \(n\) would yield valid solutions within the interval, emphasized the importance of considering both the infinite nature and the restrictions of trigonometric solutions in practical scenarios.