Problem 39 - Algebra 2 and Trigonometry

Question

Show that the graph of \(y=\sin x\) is the graph of \(y=\cos \left(x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The graphs are the same because \(\cos\left(x-\frac{\pi}{2}\right) = \sin x\) for all \(x\).

Step 1: Understand the Problem

We need to show that the functions \(y = \sin x\) and \(y = \cos\left(x - \frac{\pi}{2}\right)\) represent the same graph. This involves proving that their values are the same for any value of \(x\).

Step 2: Use the Cosine to Sine Identity

Recall the trigonometric identity that relates sine and cosine functions: \(\cos(\theta - \frac{\pi}{2}) = \sin \theta\). This identity is essential for converting the cosine expression into a sine expression.

Step 3: Apply the Identity to the Function

Substitute \(x\) for \(\theta\) in the identity: \(\cos\left(x - \frac{\pi}{2}\right) = \sin x\). This shows that \(y = \sin x\) is indeed the graph of \(y = \cos\left(x - \frac{\pi}{2}\right)\).

Step 4: Conclusion

Since \(\cos\left(x - \frac{\pi}{2}\right) = \sin x\), both functions have identical inputs and outputs for any \(x\). Therefore, the graphs of the two functions are the same.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental building blocks in trigonometry, which is the branch of mathematics that explores the relationships between the sides and angles of triangles. They are crucial for understanding wave patterns, oscillations, and circular motion.

Both functions are periodic, meaning they repeat their values in regular intervals. The sine function is typically represented by the equation:

• \( y = \sin x \)

This function produces a wave-like pattern that oscillates above and below the x-axis, with a period of \( 2\pi \). The maximum value of sine is 1, and its minimum value is -1.

The cosine function is very similar in behavior to the sine function, with its general equation:

• \( y = \cos x \)

Like sine, cosine also oscillates between -1 and 1. However, a key difference is that the cosine wave reaches its maximum value of 1 at \( x = 0 \), while the sine wave crosses through zero at this point.

Understanding these functions helps in visualizing rotational motion and analyzing electrical signals, sound waves, and light waves.

Graph Transformations

Graph transformations involve shifting, stretching, or flipping the graph of a function in various ways. These adjustments change how the graph appears, helping us identify key properties and behaviors. For trigonometric functions, transformations can express relationships between sine and cosine.

There are several types of transformations that can be applied to graphs:

• Vertical Shifts: Moving a graph up or down along the y-axis. This doesn't alter the shape, just the position.
• Horizontal Shifts: Moving a graph left or right along the x-axis, which can be done by adding or subtracting a value inside the function's argument.
• Reflections: Flipping a graph over a line, such as the x-axis, effectively changing its orientation.
• Stretching/Compressing: Altering the "width" or "height" of the wave, making it spread out or squish together.

When working with sine and cosine graphs, one common transformation is the phase shift, which is a type of horizontal shift. Recognizing and applying these transformations helps solve complex problems involving trigonometric functions, resulting in better comprehension of their applications in physics and engineering.

Phase Shift

Phase shift is a transformation applied to periodic functions, such as sine and cosine, whereby the entire graph of the function is shifted horizontally. This adjustment helps in aligning graphs or synchronizing signals, which is particularly useful in fields like signal processing and engineering.

For the cosine function, a phase shift is expressed by modifying the argument inside the function:

• \( y = \cos(x - c) \)

Here, the value \( c \) is the phase shift. If \( c > 0 \), the graph shifts to the right; if \( c < 0 \), it shifts to the left. The relationship between sine and cosine through phase shifts can be exemplified as:

• \( y = \cos(x - \frac{\pi}{2}) \) is equivalent to \( y = \sin x \)

This shows that a cosine graph can transform into a sine graph by a phase shift of \( \frac{\pi}{2} \) to the right.

Through understanding phase shifts, students can more easily manipulate and interpret trigonometric functions, improving their ability to solve engineering and physics-related problems that involve waves and oscillations.