Problem 54 - Trigonometry

Question

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \cos 4 x=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{6} + \frac{k\pi}{2}\) and \(x = \frac{\pi}{3} + \frac{k\pi}{2}\).

Step 1: Understanding the Problem

The problem asks us to find the solutions for the equation \(\cos 4x = -\frac{1}{2}\) in terms of radians, using only exact values. We need to determine the values of \(x\) for which this equation holds true.

Step 2: Use Known Cosine Values

Recall that the cosine function equals \(-\frac{1}{2}\) at certain standard angles. Specifically, \(\cos \theta = -\frac{1}{2}\) at \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\), plus multiples of \(2\pi\). Therefore, we have the general solutions for \(\theta\) as \(\frac{2\pi}{3} + 2k\pi\) and \(\frac{4\pi}{3} + 2k\pi\), for any integer \(k\).

Step 3: Solve for \(4x\)

Set \(4x = \theta\) where \(\theta\) takes the general values we found. This gives us two sets of equations: 1. \(4x = \frac{2\pi}{3} + 2k\pi\)2. \(4x = \frac{4\pi}{3} + 2k\pi\)where \(k\) is any integer.

Step 4: Divide by 4 to Solve for \(x\)

Solve each equation for \(x\) by dividing every term by 4:1. \(x = \frac{2\pi}{12} + \frac{k\pi}{2}\) simplifies to \(x = \frac{\pi}{6} + \frac{k\pi}{2}\)2. \(x = \frac{4\pi}{12} + \frac{k\pi}{2}\) simplifies to \(x = \frac{\pi}{3} + \frac{k\pi}{2}\).

Step 5: Combine the Solutions

Combine the solutions obtained from both equations. The complete solution for \(x\) is:\[ x = \frac{\pi}{6} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{\pi}{3} + \frac{k\pi}{2} \] for any integer \(k\). These represent all solutions to the original problem.

Key Concepts

Cosine Values

Cosine values play a crucial role in solving trigonometric equations like the one in the exercise. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. Beyond right triangles, the cosine function helps us determine the x-coordinate of a point on the unit circle associated with an angle in standard position.
In this exercise, we needed to find when the cosine of an angle equals \(-\frac{1}{2}\). This specific cosine value occurs at certain well-known angles, such as \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \). These angles can be deduced by understanding the symmetry and periodic nature of the cosine wave, especially in the third and second quadrants where cosine is negative.

Radian Measure

Radians provide a way to measure angles based on the arc length of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the radius of a circle to measure angles. One full circle is equal to \(2\pi\) radians.
This unit is particularly useful in calculus and other fields of mathematics because it directly relates the linear and angular dimensions. In the original exercise, all solutions are required in radians, which demand us to think in terms of arc lengths instead of degrees. When working with the equation \( \cos 4x = -\frac{1}{2} \), we recognize and find these angles in terms of radians such as \( \frac{2\pi}{3} \) or \( \frac{4\pi}{3} \).

Exact Values

Exact values in trigonometry refer to using precise and simple fractions to express angles and outcomes instead of approximate decimal numbers. They find significant applications in mathematical proofs and theoretical explorations.
In our exercise, using exact values means representing solutions in forms such as \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \), rather than decimals like 0.523598 or 1.0472. Exact values help maintain accuracy and prevent rounding errors in calculations. Therefore, finding solutions using exact values is crucial to maintain mathematical integrity, especially when these solutions are further used in more complex equations.

General Solution

The concept of a general solution is distinguished from a particular solution, as it seeks to find all values that satisfy a given equation rather than a specific instance. For trigonometric equations, solutions are periodic, meaning they repeat after certain intervals called the period.
In this exercise, once the basic solutions were found using known cosine values, they had to be expressed as a general solution that includes all possible angles that fulfill the condition \( \cos 4x = -\frac{1}{2} \) by adding multiples of the period \(2\pi\). This is accomplished by adding \(2k\pi\) to each root, where \(k\) is an integer, leading to solutions like \(x = \frac{\pi}{6} + \frac{k\pi}{2}\) and \(x = \frac{\pi}{3} + \frac{k\pi}{2}\), ensuring every potential cycle of solutions is captured.