Problem 53 Graph each of the functions by f... [FREE SOLUTION]

Chapter 5: Problem 53

Graph each of the functions by first rewriting it as a sine, cosine, or tangent of a difference or sum. $$ y=\sin x \sin \left(\frac{\pi}{4}\right)+\cos x \cos \left(\frac{\pi}{4}\right) $$

Short Answer

Expert verified

The function is $ y = \cos\left(x - \frac{\pi}{4}\right) $, a cosine wave shifted right by $ \frac{\pi}{4} $.

Step by step solution

Recognize the Trigonometric Identity

The equation given, $ y = \sin x \sin \left(\frac{\pi}{4}\right) + \cos x \cos \left(\frac{\pi}{4}\right) $, resembles the cosine sum identity. The identity is: $ \cos(a - b) = \cos a \cos b + \sin a \sin b $.

Apply the Cosine Sum Identity

Using the identity $ \cos(a - b) = \cos a \cos b + \sin a \sin b $, we equate with our function: $ a = x $ and $ b = \frac{\pi}{4}$. Therefore, $ \sin x \sin \left(\frac{\pi}{4}\right) + \cos x \cos \left(\frac{\pi}{4}\right) = \cos\left(x - \frac{\pi}{4}\right) $.

Simplify the Function

The function simplifies to $ y = \cos\left(x - \frac{\pi}{4}\right) $. This means the given function is a cosine function with a phase shift.

Graph the Function

To graph $ y = \cos\left(x - \frac{\pi}{4}\right) $, recognize it's a standard cosine wave shifted to the right by $ \frac{\pi}{4} $. The amplitude is 1, the period is $ 2\pi $, and the vertical shift is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function

The sine function is a fundamental component of trigonometry, used to describe oscillating phenomena such as sound waves, light, and the motion of pendulums. It is expressed mathematically as $ y = \sin x $, where $ x $ represents the angle in radians. This function has a few key characteristics:

**Periodicity**: It repeats every $ 2\pi $ radians. This regularity makes it a periodic function.
**Range**: The sine function oscillates between -1 and 1, giving it this specific range of values.
**Zeros**: These occur at multiples of $ \pi $, that is, when $ x = 0, \pi, 2\pi, \ldots $.

When solving problems involving the sine function, it's important to consider these properties, as they provide a framework for understanding angles and their corresponding sine values. Moreover, combining the sine function with other trigonometric functions can help solve complex equations, as seen in breaks of compounds like products into sums.

Cosine Function

The cosine function shares many similarities with the sine function, yet has distinct features that make it unique. It is represented as $ y = \cos x $. Here are some important aspects of the cosine function:

**Shape and Period**: Similar to the sine function, the cosine function is periodic with a period of $ 2\pi $. Its wave peaks at $ 1 $ when$ x = 0 $, more directly showing its maxima.
**Symmetry**: Unlike the sine function, the cosine function is symmetric about the y-axis. This property is known as even, meaning $ \cos(-x) = \cos x $.
**Values**: Maximum at $ 1 $ and minimum at $ -1 $, the cosine curve starts from $ (0, 1) $ for $ x = 0 $.

Understanding the cosine function's properties is essential for using the cosine sum and difference identities effectively. These identities enable the transformation of equations, simplifying otherwise complicated expressions, as demonstrated in the original exercise. They allow us to turn products of trigonometric functions into sums or differences, making them easier to graph or analyze.

Phase Shift

Phase shift is an important concept that shifts the wave horizontally along the x-axis without altering its shape. When we modify a trigonometric function such as cosine or sine, the term usually added or subtracted inside the function modifies its phase. This is crucial for understanding functions such as $ y = \cos(x - \frac{\pi}{4}) $:

**Direction**: A minus sign within the function $ (x - c) $ indicates a rightward shift by $ c $ units. Conversely, a plus sign $ (x + c) $ indicates a leftward shift.
**Effect on Graph**: The graph's shape remains constant, but its starting point along the x-axis shifts. In our example, shifting right by $ \frac{\pi}{4} $ moves every point in the graph accordingly.
**Practical Application**: This concept is often used to align waves in synchronization with real-world patterns, such as in signal processing or synchronizing lights in traffic systems.

Knowing how to compute and apply phase shifts is essential for accurately plotting trigonometric functions and for real-world applications where these shifts are employed to match or analyze cycles and patterns. It ensures correct placement of waves on a graph, allowing precise interpretations and analyses.

91影视

Graph each of the functions by first rewriting it as a sine, cosine, or tangent of a difference or sum. $$ y=\sin x \sin \left(\frac{\pi}{4}\right)+\cos x \cos \left(\frac{\pi}{4}\right) $$

Short Answer

Step by step solution

Recognize the Trigonometric Identity

Apply the Cosine Sum Identity

Simplify the Function

Graph the Function

Key Concepts

Sine Function

Cosine Function

Phase Shift

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