Problem 66 - Precalculus Mathematics for Calculus

Question

(a) Graph the function and make a conjecture, then (b) prove that your conjecture is true. $$y=-\frac{1}{2}[\cos (x+\pi)+\cos (x-\pi)]$$

Step-by-Step Solution

Answer

The function simplifies to \(y = \cos(x)\) and behaves as such.

Step 1: Simplify Inside the Function

First simplify the expression \(\cos(x+\pi)\) and \(\cos(x-\pi)\). Using the trigonometric identity \(\cos(\theta + \pi) = -\cos(\theta)\), we have:\[\cos(x+\pi) = -\cos(x)\]\[\cos(x-\pi) = -\cos(x)\]Thus, \[y = -\frac{1}{2}[-\cos(x) - \cos(x)] = -\frac{1}{2}[-2\cos(x)] = \cos(x)\].

Step 2: Graph the Simplified Function

Now graph the simplified function \(y = \cos(x)\). The cosine function is periodic with a period of \(2\pi\) and an amplitude of 1. The graph oscillates between 1 and -1.

Step 3: Make a Conjecture

Based on the graph of the function \(y = \cos(x)\), our conjecture is that the original function \(-\frac{1}{2}[\cos(x+\pi) + \cos(x-\pi)]\) simplifies to \(\cos(x)\) and behaves as such for all \(x\).

Step 4: Prove the Conjecture

We have already seen from Step 1 that:\[-\frac{1}{2}[\cos(x+\pi) + \cos(x-\pi)] = \cos(x)\]This was obtained by using the trigonometric identity for the cosine function shifting by \(\pi\). Thus, the conjecture is indeed proven.

Key Concepts

Graphing Functions

When graphing the function, the first step is to understand the shape and attributes of the graph you are drawing. For this exercise, the original function

• \(-\frac{1}{2}[\cos(x+\pi) + \cos(x-\pi)]\) was simplified to \(\cos(x)\)
• because of applying the trigonometric identity \(\cos(\theta + \pi) = -\cos(\theta)\).

The cosine function \(y = \cos(x)\) has a well-defined graph pattern:

• The graph oscillates in a wave-like form, known as sinusoidal.
• It has peaks, called maxima, at 1 and troughs, called minima, at -1.
• The wave repeats every \(2\pi\) units along the x-axis, which represents the period.

To create the graph:

• Identify key points at \(0, \pi/2, \pi, 3\pi/2, 2\pi\).
• Plot these points and draw a smooth curve through them.

By fully understanding each aspect of the cosine wave, you can graph many variations of this function, just by adjusting its period, amplitude, or phase shift.

Simplifying Expressions

Simplifying expressions in mathematics, especially with trigonometric identities, involves recognizing patterns and applying known rules to reduce complexity. In this exercise, the function \(-\frac{1}{2}[\cos(x+\pi) + \cos(x-\pi)]\) was simplified using trigonometric identities.The important steps include:

• Recognizing that \(\cos(x+\pi)\) equals \(-\cos(x)\), thanks to the identity \(\cos(\theta + \pi) = -\cos(\theta)\).
• Applying this identity, we find that both \(\cos(x+\pi)\) and \(\cos(x-\pi)\) simplify to \(-\cos(x)\).
• This leads to the expression \(-\frac{1}{2}[-2\cos(x)]\).
• Simplifying this further gives \(\cos(x)\).

Simplifying expressions is like solving a puzzle; seeing the right identities can make the expression much easier to handle.

Periodicity of Functions

Periodicity refers to the interval after which a function repeats its values. For trigonometric functions, periodicity is a crucial aspect to understand their graph behavior over the entire x-axis.The cosine function \(y = \cos(x)\) is known for:

• Repeating its values every \(2\pi\) units.
• Having a period of \(2\pi\), meaning the pattern of the graph exactly repeats every \(2\pi\).

This periodic nature is used in simplifying trigonometric equations and solving complex problems involving wave functions.In our exercise:

• The original function became \(\cos(x)\), which naturally inherits this periodicity.
• This means that \(y = -\frac{1}{2}[\cos(x+\pi) + \cos(x-\pi)]\) also repeats every \(2\pi\).

Understanding periodicity helps in predicting graph behaviors beyond the initial plotted range, giving insights into how functions behave across all x-values.