Problem 2 - Precalculus

Question

How does the graph of \(y=\sin x\) compare with the graph of \(y=\cos x ?\) Explain how you could horizontally translate the graph of \(y=\sin x\) to obtain \(y=\cos x .\)

Step-by-Step Solution

Answer

Translate \(y = \sin x\) left by \(\pi/2\) to match \(y = \cos x\).

Step 1: Understand the sine function

The graph of the function \(y = \sin x\) is a periodic wave that starts at the origin (0,0), goes up to 1, back to 0, down to -1, and then returns to 0, repeating every \(2\pi\) units (the period). The key points occur at \((0,0), (\pi/2, 1), (\pi, 0), (3\pi/2, -1), (2\pi, 0)\).

Step 2: Understand the cosine function

The graph of the function \(y = \cos x\) also represents a periodic wave, but it starts at (0,1) instead of (0,0) and follows the same shape as the sine function. Its key points are \((0,1), (\pi/2, 0), (\pi, -1), (3\pi/2, 0), (2\pi, 1)\).

Step 3: Recognize the relationship between sine and cosine

Notice that each corresponding point on \(y = \sin x\) is horizontally shifted from \(y = \cos x\) by \(\pi/2\) units to the left. For example, \(y = \sin(0) = 0\) aligns with \(y = \cos(\pi/2) = 0\), \(y = \sin(\pi/2) = 1\) aligns with \(y = \cos(0) = 1\), and so on.

Step 4: Describe the horizontal translation

To transform the graph of \(y = \sin x\) into \(y = \cos x\), you need to translate the sine graph \(\pi/2\) units to the left. This can be achieved by rewriting the sine function as \(y = \sin(x + \pi/2)\). This horizontal shift aligns the graphs of \(y = \sin x\) and \(y = \cos x\) perfectly.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is defined on the Cartesian coordinate system and is represented by the equation \[ y = \sin x \] Here, the input variable \(x\) typically represents an angle, and the sine function produces a corresponding value that can vary between -1 and 1. The graph of the sine function is sinusoidal, meaning that it takes the shape of smooth, continuous waves that oscillate above and below the x-axis.

• The sine wave starts at the origin (0,0) and moves up to the peak (1), falls back to the center (0), dips to the trough (-1), and returns to the center to complete one cycle.
• The distance over which the wave completes one full cycle is referred to as the period, which for the sine function is \(2\pi\).
• Important points include:
  • Start point: (0,0)
  • First peak: \((\pi/2, 1)\)
  • Center return: \((\pi, 0)\)
  • Lowest point: \((3\pi/2, -1)\)
  • End of first cycle: \((2\pi, 0)\)

Understanding these points helps in drawing or identifying the graph of the sine function accurately.

Cosine Function

The cosine function, like the sine function, is a crucial trigonometric function. The cosine function graph is similar to the sine graph, but shifted horizontally. The equation that describes the cosine function is \[ y = \cos x \] The cosine function produces values based on the x-coordinate of a point on the unit circle.

• The cosine wave starts at its maximum point at \(y = 1\) when \(x=0\).
• It then follows the wave pattern: descending to 0, dropping to its minimum (-1), rising back to 0, and returning to its maximum (1) to complete a full cycle.
• Just like the sine, its period is also \(2\pi\), representing the horizontal length of one complete wave cycle.
• Key points to remember:
  • Maximum start: \((0,1)\)
  • Zero crossing: \((\pi/2, 0)\)
  • Minimum: \((\pi, -1)\)
  • Second zero crossing: \((3\pi/2, 0)\)
  • Maximum return: \((2\pi, 1)\)

Recognizing these points is key to understanding how the cosine function behaves.

Horizontal Translation

Horizontal translation refers to the shifting of a graph along the x-axis in the Cartesian plane. When it comes to the sine and cosine functions, this concept explains how these graphs are essentially the same shape but offset by a certain amount along the x-axis.In mathematical terms, translating a function horizontally means adjusting the input value within the function. Specifically, translating the \[ y = \sin x \]to become the graph of \[ y = \cos x \]requires moving the sine wave \(\pi/2\) units to the left:

• This shift means that each point on the sine wave aligns with the corresponding point on the cosine wave.
• Mathematically, this is represented by modifying the sine function to \[ y = \sin(x + \pi/2) \]
• This adjustment ensures the graphs coincide: the peak at \((\pi/2, 1)\) on the sine graph, for instance, aligns perfectly with the cosine's peak at \((0, 1)\).

Understanding horizontal translations helps in visualizing and manipulating trigonometric functions, allowing for greater flexibility in analyzing periodic patterns.