Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \cos \left(4 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2} $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{8} + \frac{k\pi}{2}\) and \(x = \frac{(1+k)\pi}{2}\) for integer \(k\).

Step 1: Recognize Special Angles

Recall that \(\cos \theta = \frac{\sqrt{2}}{2}\) when \(\theta = \frac{\pi}{4}\) or \(\theta = \frac{7\pi}{4}\). These are the angles in the first and fourth quadrants where the cosine value is \(\frac{\sqrt{2}}{2}\).

Step 2: Set Up Equations

Set up the initial equations based on the known cosine values: \(4x - \frac{\pi}{4} = \frac{\pi}{4} + 2k\pi\) and \(4x - \frac{\pi}{4} = \frac{7\pi}{4} + 2k\pi\), where \(k\) is any integer representing additional revolutions.

Step 3: Solve the First Equation

Starting with the first equation: \(4x - \frac{\pi}{4} = \frac{\pi}{4} + 2k\pi\). Add \(\frac{\pi}{4}\) to both sides to get: \(4x = \frac{\pi}{2} + 2k\pi\). Then, divide everything by 4: \(x = \frac{\pi}{8} + \frac{k\pi}{2}\).

Step 4: Solve the Second Equation

Now solve the second equation: \(4x - \frac{\pi}{4} = \frac{7\pi}{4} + 2k\pi\). Add \(\frac{\pi}{4}\) to both sides to obtain: \(4x = 2\pi + 2k\pi\), which simplifies to \(4x = 2(1+k)\pi\). Divide everything by 4: \(x = \frac{(1+k)\pi}{2}\).

Step 5: General Solution

Based on both solutions, we gather that the general solutions are given by \(x = \frac{\pi}{8} + \frac{k\pi}{2}\) and \(x = \frac{(1+k)\pi}{2}\), where \(k\) is any integer.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the unit circle, which is a circle with a radius of one centered at the origin, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.
This means that for an angle \(\theta\), \(\cos \theta\) corresponds to this x-coordinate. The cosine function repeats its values in a cyclical pattern, completing one full cycle every \(2\pi\) radians or 360 degrees, which is known as the periodicity of the cosine function.

• Cosine values range from -1 to 1.
• It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• It has a cycle repetition every \(2\pi\).

Special Angles

Special angles are those angles for which the trigonometric functions yield simple, exact values. For the cosine function, some of these special angles include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},\) and \(\frac{\pi}{2}\). Each of these angles can be found using the circle geometry and corresponding symmetries.
In our example problem, we utilize the special angles \(\frac{\pi}{4}\) and \(\frac{7\pi}{4}\), both of which have a cosine value of \(\frac{\sqrt{2}}{2}\).

• Special angles like \(\frac{\pi}{4}\) help in solving trigonometric equations.
• These angles are used to identify specific, familiar values of trigonometric functions without a calculator.
• In quadrant analysis, \(\frac{\pi}{4}\) is found in the first quadrant; \(\frac{7\pi}{4}\) is in the fourth quadrant.

General Solutions

When solving trigonometric equations, it's about more than finding immediate solutions; it's also about identifying all the potential solutions. This involves the concept of general solutions, where you find all possible angles that satisfy the equation.
Due to the periodic nature of the cosine function, these solutions are found by adding integer multiples of the function's period \(2\pi\). For the exercise at hand, the general solutions can be summarized as:

• For \(4x - \frac{\pi}{4} = \frac{\pi}{4} + 2k\pi\), solve to get \(x = \frac{\pi}{8} + \frac{k\pi}{2}\).
• For \(4x - \frac{\pi}{4} = \frac{7\pi}{4} + 2k\pi\), solve to find \(x = \frac{(1+k)\pi}{2}\).
• The \(k\) adjusts the angle based on how many full cycles (or periods) have been completed, to capture all solutions.

Understanding general solutions ensures you can capture every possible angle solution for a trigonometric equation.