Problem 52 - Precalculus Mathematics for Calculus

Question

Solve the given equation. $$\sec \theta(2 \cos \theta-\sqrt{2})=0$$

Step-by-Step Solution

Answer

Cosine equals \(\frac{\sqrt{2}}{2}\) at \(\theta = \frac{\pi}{4} + 2k\pi\) and \(\theta = \frac{7\pi}{4} + 2k\pi\).

Step 1: Identify the Zero Product Property

The equation is \[\sec \theta (2 \cos \theta - \sqrt{2}) = 0\]According to the zero product property, if the product of two terms is zero, at least one of the terms must be zero.

Step 2: Solve for Secant Factor

The first term to consider is \(\sec \theta = 0\). However, the secant function \( \sec \theta = \frac{1}{\cos \theta}\) is never zero, since there is no angle \(\theta\) for which \(\cos \theta = \infty\). Therefore, \(\sec \theta = 0\) has no valid solutions.

Step 3: Solve for Cosine Factor

The second term to consider is \[2 \cos \theta - \sqrt{2} = 0\]Solve for \(\cos \theta\):\[2 \cos \theta = \sqrt{2}\]\[\cos \theta = \frac{\sqrt{2}}{2}\]

Step 4: Identify Angles for Cosine

The cosine function equals \(\frac{\sqrt{2}}{2}\) at specific angles. These angles are:\[\theta = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad \theta = \frac{7\pi}{4} + 2k\pi\]where \(k\) is any integer, representing the periodic nature of the cosine function.

Key Concepts

Zero Product Property

The Zero Product Property states that if a product of two terms equals zero, at least one of the terms must be zero. It's a powerful tool in solving equations like the one in the exercise. When you see something like \( a \times b = 0 \), either \( a = 0 \) or \( b = 0 \), or both. This property simplifies multi-factor equations, allowing you to investigate each factor independently. For our equation \( \sec \theta(2 \cos \theta - \sqrt{2}) = 0 \), the Zero Product Property suggests that either \( \sec \theta = 0 \) or \( 2 \cos \theta - \sqrt{2} = 0 \). This approach significantly narrows down the potential solutions by focusing on simpler parts of the problem. By applying this concept, you efficiently evaluate each factor separately to explore valid solutions.

Secant Function

The secant function, written as \( \sec \theta \), is one of the six primary trigonometric functions. Defined as the reciprocal of the cosine function, the equation is: \( \sec \theta = \frac{1}{\cos \theta} \). Unlike sine and cosine, which range from -1 to 1, the secant function has values greater than or equal to 1 or less than or equal to -1, except where it's undefined.
Since \( \sec \theta \) is the reciprocal of \( \cos \theta \), it becomes undefined when \( \cos \theta = 0 \). In terms of the original problem, \( \sec \theta = 0 \) has no solutions because no real number makes the cosine infinity. Since the secant divides one by the cosine, it can never be zero itself. This realization shows that the equation's solution must come from the other factor of the equation.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions that describe the x-coordinate of a point on the unit circle. It shows the horizontal distance or position of an angle, making it critical in many mathematical applications. In the equation \( 2 \cos \theta - \sqrt{2} = 0 \), we are tasked with finding values for \( \theta \) that satisfy \( \cos \theta = \frac{\sqrt{2}}{2} \).

• The cosine function reaches \( \frac{\sqrt{2}}{2} \) at specific angles, specifically \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \) within the first full circle of the unit circle.
• These angles originate from the 45-degree angles where both sine and cosine share the same values.

Understanding these key angles is crucial for identifying solutions when the cosine value is specified. This trigonometric interplay with the unit circle supports a wide range of mathematical principles essential for solving equations in trigonometry.

Periodic Nature of Trigonometric Functions

Trigonometric functions, like sine and cosine, are periodic in nature, meaning they repeat their values at regular intervals. This periodicity is vital to solving trigonometric equations because it shows that each function is cyclical with consistent repetitive behaviors. The cosine function, in particular, has a period of \( 2\pi \), repeating every full rotation in the unit circle.
In our problem, identifying \( \theta \) values when \( \cos \theta = \frac{\sqrt{2}}{2} \) gives \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \). Because of periodicity, these solutions are generalizable to multiple rotations around the circle with the addition of \( 2k\pi \), where \( k \) is any integer. This concept is useful in finding the complete set of solutions to trigonometric equations within any given domain by acknowledging the consistent repeating nature of these functions.