Problem 51 - Algebra and Trigonometry

Question

\(51-52\) (a) Graph the function and make a conjecture, then (b) prove that your conjecture is true. $$ y=\sin ^{2}\left(x+\frac{\pi}{4}\right)+\sin ^{2}\left(x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The function is constant at \( y = 1 \).

Step 1: Identify the Function Components

The given function is composed of two terms: \( \sin^2\left(x + \frac{\pi}{4}\right) \) and \( \sin^2\left(x - \frac{\pi}{4}\right) \). Each term involves squaring a shifted sine function.

Step 2: Graph the Function

To graph the function \( y = \sin^2\left(x + \frac{\pi}{4}\right) + \sin^2\left(x - \frac{\pi}{4}\right) \), plot it over one period of the sine function. Based on the periodic nature of sine, the graph repeats itself every \( 2\pi \) units. Use graphing software or a graphing calculator for visual observation. Observe that the graph has a constant value.

Step 3: Make a Conjecture

From the graph, it appears that the function remains constant throughout. The conjecture is that \( y = 1 \) for all values of \( x \). This suggests the possibility of the function being simplified to a constant value.

Step 4: Simplify Using Trigonometric Identities

Recall the trigonometric identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \). Use this to express each sine squared term: \[ \sin^2\left(x + \frac{\pi}{4}\right) = \frac{1 - \cos\left(2x + \frac{\pi}{2}\right)}{2} \] and \[ \sin^2\left(x - \frac{\pi}{4}\right) = \frac{1 - \cos\left(2x - \frac{\pi}{2}\right)}{2} \].

Step 5: Simplify Further

Using the fact that \( \cos\left(2x + \frac{\pi}{2}\right) = -\sin(2x) \) and \( \cos\left(2x - \frac{\pi}{2}\right) = \sin(2x) \), both expressions become: \[ \sin^2\left(x + \frac{\pi}{4}\right) + \sin^2\left(x - \frac{\pi}{4}\right) = \frac{1 - (-\sin(2x))}{2} + \frac{1 - \sin(2x)}{2} \].

Step 6: Combine and Conclude

Simplify the combined expression: \[ = \frac{1 + \sin(2x) + 1 - \sin(2x)}{2} = \frac{2}{2} = 1 \]. Thus, the function reduces to \( y = 1 \) which confirms the conjecture made from the graph.

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the fundamental functions in trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The sine function is periodic with a period of \( 2\pi \), meaning that its pattern repeats every \( 2\pi \) units. This characteristic makes it extremely useful in modeling wave-like phenomena, among other things.

In this exercise, we deal with expressions such as \( \sin^2(x + \frac{\pi}{4}) \), which indicate that the sine function is not only taken at shifted values of \( x \) but also squared. Squaring a sine value results in all outputs being positive or zero since \( (-a)^2 = a^2 \) for any \( a \). Thus, \( \sin^2(x) \) will always lie between 0 and 1.

Understanding how these manipulations affect the graph of the function is essential for analyzing and simplifying trigonometric expressions effectively.

Graphing Trigonometric Functions

Graphing trigonometric functions like the sine function helps to visualize their behavior over a period. When graphing a function like \( y = \sin^2(x + \frac{\pi}{4}) + \sin^2(x - \frac{\pi}{4}) \), it's useful to plot the graph over one full period, which for \( \sin(x) \) is \( 0 \) to \( 2\pi \).

Each shift in the function, such as \( \frac{\pi}{4} \), adjusts the starting point of the sine curve along the x-axis. Since both components of this function are squared, their individual fluctuations add up. After using trigonometric identities, we discover that the graph of this particular function results in a constant value over the entire x-axis. This illustrates the power of combining algebraic manipulation with graphical interpretation to understand complex functions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables and are pivotal in simplifying trigonometric expressions. For this problem, one of the key identities used is the Pythagorean identity expressed as \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \). This identity helps transform and simplify \( \sin^2 \) expressions by connecting them to cosine functions.

In step 5 of the solution, simplifying the cosine terms such as \( \cos(2x + \frac{\pi}{2}) = -\sin(2x) \) and \( \cos(2x - \frac{\pi}{2}) = \sin(2x) \) is crucial in the progression to the conjectured constant value. Recognizing and applying these identities facilitates the transformation of a seemingly complex equation to a much simpler form, \( y = 1 \), confirming that the function remains constant for any \( x \). These insights underscore the utility of trigonometric identities in both mathematical theory and resolving practical problems.