Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b) \(y=\cos \left(x+\frac{\pi}{2}\right)\) (c) \(y=-\sin (x+\pi)\) (d) \(y=\cos (x-\pi)\) (e) \(y=-\sin (\pi-x)\) (f) \(y=\cos \left(x-\frac{\pi}{2}\right)\) (g) \(y=-\cos (\pi-x)\) (h) \(y=\sin \left(x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Functions (a) & (g); (b) & (e); (c) & (f); (d) & (h) represent the same graphs.

Step 1: Analyze Function (a)

The function is given as \( y=\sin \left(x+\frac{\pi}{2}\right) \). Using the identity \( \sin(x + \frac{\pi}{2}) = \cos(x) \), the function simplifies to \( y = \cos(x) \).

Step 2: Analyze Function (b)

The function is \( y=\cos \left(x+\frac{\pi}{2}\right) \). Using the identity \( \cos(x + \frac{\pi}{2}) = -\sin(x) \), the function simplifies to \( y = -\sin(x) \).

Step 3: Analyze Function (c)

The function is \( y=-\sin (x+\pi) \). Using the identity \( \sin(x + \pi) = -\sin(x) \), the function simplifies to \( y = -(-\sin(x)) = \sin(x) \).

Step 4: Analyze Function (d)

The function is \( y=\cos (x-\pi) \). Using the identity \( \cos(x - \pi) = -\cos(x) \), the function simplifies to \( y = -\cos(x) \).

Step 5: Analyze Function (e)

The function is \( y=-\sin (\pi-x) \). Using the identity \( \sin(\pi-x) = \sin(x) \), the function simplifies to \( y = -\sin(x) \).

Step 6: Analyze Function (f)

The function is \( y=\cos \left(x-\frac{\pi}{2}\right) \). Using the identity \( \cos(x - \frac{\pi}{2}) = \sin(x) \), the function simplifies to \( y = \sin(x) \).

Step 7: Analyze Function (g)

The function is \( y=-\cos (\pi-x) \). Using the identity \( \cos(\pi-x) = -\cos(x) \), the function simplifies to \( y = -(-\cos(x)) = \cos(x) \).

Step 8: Analyze Function (h)

The function is \( y=\sin \left(x-\frac{\pi}{2}\right) \). Using the identity \( \sin(x - \frac{\pi}{2}) = -\cos(x) \), the function simplifies to \( y = -\cos(x) \).

Step 9: Group Functions with Same Simplification

- \( y = \cos(x) \) is represented by functions from (a) and (g).- \( y = -\sin(x) \) is represented by functions from (b) and (e).- \( y = \sin(x) \) is represented by functions from (c) and (f).- \( y = -\cos(x) \) is represented by functions from (d) and (h).

Key Concepts

Sine Function

The sine function, represented as \( y = \sin(x) \), is a fundamental trigonometric function that describes a smooth, repetitive oscillation. It mimics the wave patterns found in nature, appearing in phenomena like sound waves and tide motions. The essential characteristics of the sine function include its amplitude, period, and phase shift, crucial for understanding its behavior on the graph.

• **Amplitude**: This is the height from the centerline to the peak of the wave. For basic \( y = \sin(x) \), the amplitude is 1.
• **Period**: This is the length of one complete wave cycle. Sine's period is \( 2\pi \).

The sine function is also known for its symmetry and specific periodic values:

• \( \sin(0) = 0 \)
• \( \sin(\pi/2) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(2\pi) = 0 \)

Understanding these points helps in comprehending more complex transformations of the sine function.

Cosine Function

The cosine function, expressed as \( y = \cos(x) \), is closely related to the sine function and shares many of its properties but shifted in phase. It also exhibits a wave-like pattern, essential for modeling various cyclic phenomena. Here are key features of the cosine function:

• **Amplitude**: Like sine, the amplitude of the cosine function is also 1.
• **Period**: The function repeats every \( 2\pi \) units.
• **Key Values**: \( \cos(0) = 1 \), \( \cos(\pi/2) = 0 \), \( \cos(\pi) = -1 \), \( \cos(3\pi/2) = 0 \), and \( \cos(2\pi) = 1 \).

A notable aspect of the cosine function is its relationship with the sine function. It can be seen as a sine wave shifted by \( \pi/2 \) to the left. This relationship can be utilized to transform between the two functions for analyzing graphs.

Graph Transformation

Transformations of sine and cosine functions involve altering their graphs to accommodate changes in properties such as amplitude, period, and phase shift. These transformations allow us to model different wave behaviors using basic trigonometric functions.

Some typical transformations include:

• **Amplitude Change**: Adjusts the height of the graph, e.g., \( a \sin(x) \) stretches the graph vertically by a factor of \( a \).
• **Period Change**: Modifies how wide the graph appears. For instance, \( \sin(bx) \) compresses or stretches the graph horizontally, altering the period to \( \frac{2\pi}{b} \).
• **Vertical Shift**: Moves the graph up or down along the y-axis. The expression \( \sin(x) + c \) shifts it upward by \( c \).

Each transformation has distinct effects on the graph's appearance, allowing for precise adjustments to wave representations.

Phase Shift

Phase shift in trigonometric functions refers to the horizontal shift left or right on a graph. It results from adding or subtracting a value within the argument of the sine or cosine function, such as in \( y = \sin(x + c) \).

• **Positive Phase Shift**: A value added inside the function's argument, shifts the graph to the left.
• **Negative Phase Shift**: Subtracting a value results in a right shift.

Phase shifts are important in functions representing oscillatory phenomena, allowing precise control over the starting position of waves. By using the identities, for example, \( \sin(x + \frac{\pi}{2}) = \cos(x) \), we can understand the transformation of a sine wave into a cosine wave.
This transformation showcases how understanding phase shifts can simplify complex wave expressions by correlating them to standard sine and cosine functions. Recognizing these transformations in equations like those in the given exercise helps in comparing and grouping equivalent trigonometric graphs effectively.