Problem 87 - Precalculus with Limits

Question

In Exercises 85 - 88, use a graphing utility to approximate the solutions in the interval \( \left[0,2\pi\right) \). \( \sin\left(x + \dfrac{\pi}{2}\right) + \cos^2 x = 0 \)

Step-by-Step Solution

Answer

The approximate solutions to the equation \( \sin\left(x + \dfrac{\pi}{2}\right) + \cos^2 x = 0 \) in the interval \( \left[0,2\pi\right) \) are \( x = \dfrac{\pi}{2} \), \( x = \pi \), and \( x = \dfrac{3\pi}{2} \).

Step 1: Simplify the equation

The given equation is \( \sin\left(x + \dfrac{\pi}{2}\right) + \cos^2 x = 0 \). Using the sine function shift property, we know that \( \sin\left(x + \dfrac{\pi}{2}\right) \) is equivalent to \( \cos x \). Substitute this into the equation to obtain \( \cos x + \cos^2 x = 0 \).

Step 2: Factor the equation

The equation from Step 1: \( \cos x + \cos^2 x = 0 \) can be factored to \( \cos x (1 + \cos x) = 0 \). According to the zero-product property, if a product is zero then at least one of its factors must be zero.

Step 3: Solve for \(x\)

Setting each factor equal to zero gives \( \cos x = 0 \) and \( 1 + \cos x = 0 \). We then solve for \( x \) in each case. For the first equation, \( x = \dfrac{\pi}{2} \) or \( x = \dfrac{3\pi}{2} \). For the second equation, \( x = \pi \). All solutions are within the interval \( \left[0,2\pi\right) \).

Step 4: Confirm solutions using a graphing utility

Plot the equation using a graphing utility such as Desmos, GeoGebra, or a graphing calculator. The x-intercepts should confirm the solutions derived from Step 3. In this case, the plot will have x-intercepts at \( x = \dfrac{\pi}{2} \), \( x = \pi \), and \( x = \dfrac{3\pi}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express one trigonometric function in terms of another. These fundamental tools allow us to simplify and solve trigonometric equations by transforming and combining them into a more workable form. One such identity of crucial importance is the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), which indicates the inherent relationship between the sine and cosine functions.

In the given exercise, the identity \( \sin(x + \frac{\pi}{2}) = \cos x \) is employed, which is rooted in the concept that the sine function is equal to the cosine of a complementary angle. This identity assists greatly in simplifying the initial equation. Understanding these identities is not just about memorizing formulas but also about recognizing patterns and relationships that can optimize the solving process.

Solving Trigonometric Equations

Trigonometric equations often require a different approach compared to algebraic equations. The key to solving them lies in utilizing trigonometric identities, factoring, and inverse trigonometric functions. When faced with a trigonometric equation, such as \( \cos x + \cos^2 x = 0 \), one effective strategy is to first transform and simplify the equation using identities.

Once simplified, we can then apply algebraic techniques such as factoring. It's essential to remember the periodic nature of trigonometric functions and that solutions may come in multiple angles that satisfy the equation within a specified interval. Identifying and verifying all possible solutions within the given interval is a critical step to ensure accuracy in the problem-solving process.

Zero-Product Property

The zero-product property is a fundamental principle in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. Utilizing this property is especially handy when dealing with factored equations.

For instance, in the equation \( \cos x (1 + \cos x) = 0 \), we set each factor equal to zero separately: \( \cos x = 0 \) and \( 1 + \cos x = 0 \). By finding the zeros of these factors, we can determine the values of \( x \) that satisfy the original equation. This property aids in unraveling the solutions that might otherwise be obscured in the format of a complex trigonometric equation.

Graphing Utility

A graphing utility is a powerful tool for visualizing and solving equations, including trigonometric ones. Modern graphing utilities like Desmos and GeoGebra offer an interactive platform where equations can be plotted quickly, revealing their graphs and key features like intercepts, maximums, and minimums.

In our trigonometric problem, after isolating the solutions algebraically, a graphing utility can be used to confirm these solutions by plotting the original equation and observing the x-intercepts. These intercepts should align with the solutions found analytically. This step not only confirms the solutions but also provides a visual comprehension of the behavior of the trigonometric function over the desired interval, enhancing understanding and reinforcing the connection between the algebraic and graphical representations.