Problem 1 - Precalculus Essentials

Question

Use substitution to determine whether the given \(x\) -value is a solution of the equation. $$\cos x=\frac{\sqrt{2}}{2}, \quad x=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

Since the left side of the equation \(\cos \left( \frac{\pi}{4} \right)\) is equal to the right side of the equation \(\frac{\sqrt{2}}{2}\), this means x=\(\frac{\pi}{4}\) is indeed a solution to the equation. So the answer is yes.

Step 1: Understand The Given Equation

The given equation is \(\cos x=\frac{\sqrt{2}}{2}\), and the given x-value is \(\frac{\pi}{4}\). The goal is to substitute this x-value into the cosine function of the equation.

Step 2: Substitute The X-value

Substitute the given x-value which is \(\frac{\pi}{4}\) into the equation. This gives:\[\cos \left( \frac{\pi}{4} \right)\]

Step 3: Calculate The Cosine

Calculate the cosine of \(\frac{\pi}{4}\). This is done by recalling the trigonometric values for basic angles. The cosine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\) due to standard trigonometric values table.

Step 4: Compare The Left Side With The Right Side

Compare the calculated value of \(\cos \left( \frac{\pi}{4} \right)\), which is \(\frac{\sqrt{2}}{2}\), with the right side of the original equation, which is also \(\frac{\sqrt{2}}{2}\).

Key Concepts

Substitution Method

In mathematics, the substitution method is a fundamental technique applied across various branches, including algebra and calculus. This method is particularly useful for simplifying complex problems by replacing variables with known values. When dealing with trigonometric equations, substitution allows us to verify if a certain angle is a solution by inserting it directly into the equation.

For instance, to determine if \( x = \frac{\pi}{4} \) is a solution for the equation \( \cos x = \frac{\sqrt{2}}{2} \), we substitute \( \frac{\pi}{4} \) in place of \( x \) and evaluate the function. The effectiveness of this method hinges on a clear understanding of the function and the known values being substituted. By directly seeing the results of the substitution, students can quickly assess if they have found a valid solution.

Trigonometric Identities

Trigonometric identities are equations that hold true for all possible values of the variable involved and are an indispensable tool in trigonometry. They help simplify equations, convert expressions, and establish relationships between trigonometric functions. Understanding these identities is vital for working with any trigonometric equation.

Common identities include reciprocal identities, Pythagorean identities, and angle sum and difference identities. When verifying solutions, such as in the given exercise, it's essential to know, for example, that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) by recalling the exact values for cosine at commonly used angles. This knowledge, coupled with methodical substitution, allows students to confirm or disprove potential solutions to trigonometric equations. Mastering these identities is a key step in successfully navigating through more complex trigonometric problems.

Cosine Function

The cosine function is one of the primary trigonometric functions and it describes the relationship between the angle of a right-angled triangle and the ratio of the adjacent side to the hypotenuse. It's defined for all real numbers and its value ranges between -1 and 1.

Understanding the cosine function is critical because it frequently appears in various forms of mathematical problems and real-life applications like signal processing, physics, and engineering. The cosine of commonly known angles such as \( \frac{\pi}{4} \) are often memorized and used directly in calculations. Recognizing that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) is part of this understanding and enables a quick evaluation of trigonometric expressions in which they appear. Grasping the behavior of the cosine function over its domain helps students predict its values and solve equations more efficiently.