Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\cos ^{2} x-\sin ^{2} x=1$$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = \pi \).

Step 1: Recognize the Identity

The equation \( \cos^2 x - \sin^2 x = 1 \) resembles a trigonometric identity. Recall the identity \( \cos 2x = \cos^2 x - \sin^2 x \). This allows us to rewrite the given equation in terms of \( \cos 2x \).

Step 2: Rewrite the Equation using Identity

Using the identity from Step 1, rewrite the equation as \( \cos 2x = 1 \). This simplifies the problem to finding the value of \( x \) where \( \cos 2x = 1 \).

Step 3: Solve for \( 2x \)

The cosine function equals 1 at \( 0, 2\pi, 4\pi, \ldots \). For the equation \( \cos 2x = 1 \), this means \( 2x = 0 + 2k\pi \) where \( k \) is an integer.

Step 4: Solve for \( x \)

Divide the equation \( 2x = 2k\pi \) by 2 to find \( x = k\pi \).

Step 5: Determine Solutions in the Interval

Considering the interval \([0, 2\pi)\), determine the integer values of \( k \) such that \( x = k\pi \) remains within the interval. The possible values are \( k = 0 \) and \( k = 1 \).

Step 6: Final Solution

Substitute \( k = 0 \) and \( k = 1 \) into \( x = k\pi \) to find \( x = 0 \) and \( x = \pi \). These are the solutions within the specified interval.

Key Concepts

Cosine Identity

Trigonometric identities are vital in solving equations involving trigonometric functions. One such identity is the cosine identity, which states:

• \[\cos 2x = \cos^2 x - \sin^2 x\]

This identity helps simplify expressions that involve squared trigonometric functions. By recognizing the identity, we can transform the given expression into a simpler form. In the original exercise, the equation \( \cos^2 x - \sin^2 x = 1 \) was converted into \( \cos 2x = 1 \) using the cosine identity. This transformation makes it easier to identify the values of \( x \) that satisfy the equation. Understanding and applying the cosine identity allows for a more straightforward approach to solving trigonometric equations.

Interval Notation

Interval notation is a way of representing a set of numbers that fall within a specified range. When solving trigonometric equations, it's crucial to know which solutions are valid within a given range. In this case, the interval \[[0, 2\pi)\] represents all the angles from 0 to just before \(2\pi\). The brackets indicate whether endpoints are included:

• The "closed" square bracket \([\) means the endpoint is included.
• The "open" parenthesis \()\) means the endpoint is not included.

Thus, the interval \([0, 2\pi)\) includes 0 but does not include \(2\pi\). When you find solutions to the trigonometric equation, it's essential to verify if they fit within this specific interval. If a solution like \(2\pi\) appears, it would not be considered a valid solution within the interval.

Trigonometric Identities

Trigonometric identities are equations relating the various trigonometric functions to each other. They are based on fundamental concepts in trigonometry and can simplify complex expressions. Common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Double Angle Identities: \( \cos 2x = \cos^2 x - \sin^2 x \) and \( \sin 2x = 2 \sin x \cos x \)
• Addition Formulas: \( \cos(x + y) = \cos x \cos y - \sin x \sin y \)

These identities form the backbone of many solution methods in trigonometry. By using them, you can transform and simplify equations, making it easier to solve for unknown values. Mastery of these identities allows for enhanced problem-solving techniques, efficiently finding solutions for trigonometric functions.

Solution Methods in Trigonometry

Solving trigonometric equations typically involves various methods. Some common methods include:

• Substitution of Identities: Use known identities to rewrite the equation.
• Graphical Methods: Graph the function to identify solutions.
• Algebraic Manipulation: Rearrange and factor expressions to isolate variables.

In the original exercise, using substitution of identities was the best approach. By recognizing that \( \cos^2 x - \sin^2 x \) is an identity, the problem was simplified to solving \( \cos 2x = 1 \). After rewriting the equation, the solution becomes straightforward, finding values where the cosine function is 1. The key to successful solution methods is selecting the approach that simplifies the process effectively, ensuring that solutions obtained fit within any specified intervals.