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Magazine Chapter 7: Problem 51
(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true. $$y=\sin ^{2}\left(x+\frac{\pi}{4}\right)+\sin ^{2}\left(x-\frac{\pi}{4}\right)$$
Short Answer
Expert verified
The function \( y = 1 \) is constant for all \( x \).
Step by step solution
01
Understanding the Function
The function given is \( y = \sin^2 \left( x + \frac{\pi}{4} \right) + \sin^2 \left( x - \frac{\pi}{4} \right) \). This involves two sine squared terms shifted by \( \frac{\pi}{4} \) in opposite directions.
02
Graph the Function
Use a graphing calculator or software to plot the function \( y = \sin^2 \left( x + \frac{\pi}{4} \right) + \sin^2 \left( x - \frac{\pi}{4} \right) \). Observe the shape and periodicity of the function. The graph should appear as a sinusoidal wave with a consistent amplitude and periodicity.
03
Make a Conjecture
Upon graphing, you should notice that the function seems to have a constant value. This suggests that \( y \) might be independent of \( x \). A reasonable conjecture is that \( y = 1 \) for all \( x \).
04
Use Trigonometric Identities for Proving
Apply trigonometric identities to simplify the function. Recall the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \). Substitute: \( y = \frac{1 - \cos(2(x + \frac{\pi}{4}))}{2} + \frac{1 - \cos(2(x - \frac{\pi}{4}))}{2} \).
05
Simplify the Expression
Substitute and expand: \( y = \frac{1 - \cos(2x + \frac{\pi}{2})}{2} + \frac{1 - \cos(2x - \frac{\pi}{2})}{2} \). Simplify using \( \cos\left(\theta + \frac{\pi}{2}\right) = -\sin(\theta) \) and \( \cos\left(\theta - \frac{\pi}{2}\right) = \sin(\theta) \), resulting in \( y = \frac{1 - (-\sin 2x)}{2} + \frac{1 - \sin 2x}{2} \).
06
Final Simplification
Combine terms in the expression: \( y = \frac{1 + \sin 2x}{2} + \frac{1 - \sin 2x}{2} \). The terms involving \( \sin 2x \) cancel each other out, leaving \( y = \frac{2}{2} = 1 \). Thus, the function simplifies to \( y = 1 \) for all \( x \).
07
Conclusion
The conjecture that \( y = 1 \) for all \( x \) is confirmed. The proof using trigonometric identities validates the graph's indication that the function is constant across all values of \( x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Functions
Graphing functions is a key skill in mathematics. By transforming algebraic expressions into visual representations, you can see how the function behaves across different values of its variable. For the function \( y = \sin^2 \left( x + \frac{\pi}{4} \right) + \sin^2 \left( x - \frac{\pi}{4} \right) \), graphing helps us see its periodic nature. This function is composed of two shifted sine waves. To begin graphing:
- Use a graphing tool.
- Input the function as it is.
- Examine the wave's height and frequency.
Sinusoidal Waves
Sinusoidal waves are waves whose graphs have the shape of a sine or cosine function. They are periodic, meaning they repeat their values in regular intervals. In the function \( y = \sin^2 \left( x + \frac{\pi}{4} \right) + \sin^2 \left( x - \frac{\pi}{4} \right) \), we encounter two sine waves that are squared. Squaring a sine wave affects its graph by:
- Altering the wave鈥檚 range 鈥� it moves from being negative to always positive as sine squared produces values between 0 and 1.
- Causing flattening 鈥� the zeros and peaks become less pronounced.
Simplifying Expressions
Simplifying trigonometric expressions often involves applying identities to reduce complexity and find underlying properties. In this exercise, simplifying \( y = \sin^2 \left( x + \frac{\pi}{4} \right) + \sin^2 \left( x - \frac{\pi}{4} \right) \) revealed a constant function. The process includes:
- Using the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \) to break down each sine squared part into manageable forms.
- Substituting and expanding these identities to expose underlying relationships, such as the effect of combining the components.
- Simplifying complex fractions and terms results in observing key cancellations (e.g., terms involving sine values cancel out between additions).
