Graphs of the Sine and Cosine Functions: Videos & Practice Problems

Understanding the sine and cosine functions is essential in trigonometry, especially when it comes to graphing these functions on a standard x-y coordinate system. The sine function corresponds to the y-coordinates of points on the unit circle, while the cosine function relates to the x-coordinates. As you traverse the unit circle, the values of sine and cosine repeat, leading to a periodic wave-like behavior in their graphs.

To graph the sine function, denoted as \( y = \sin(x) \), you can start by plotting key points based on the unit circle. For instance, at \( x = 0 \), \( \sin(0) = 0 \); at \( x = \frac{\pi}{2} \), \( \sin\left(\frac{\pi}{2}\right) = 1 \); at \( x = \pi \), \( \sin(\pi) = 0 \); at \( x = \frac{3\pi}{2} \), \( \sin\left(\frac{3\pi}{2}\right) = -1 \); and at \( x = 2\pi \), \( \sin(2\pi) = 0 \). Connecting these points reveals a smooth wave pattern that continues indefinitely due to the periodic nature of the sine function.

Similarly, the cosine function, represented as \( y = \cos(x) \), can be graphed using its corresponding values. At \( x = 0 \), \( \cos(0) = 1 \); at \( x = \frac{\pi}{2} \), \( \cos\left(\frac{\pi}{2}\right) = 0 \); at \( x = \pi \), \( \cos(\pi) = -1 \); at \( x = \frac{3\pi}{2} \), \( \cos\left(\frac{3\pi}{2}\right) = 0 \); and at \( x = 2\pi \), \( \cos(2\pi) = 1 \). The cosine graph also exhibits a wave pattern, but it starts at a maximum value of 1, contrasting with the sine graph that begins at 0.

In both graphs, the highest points are referred to as crests or peaks, while the lowest points are known as troughs or valleys. The amplitude of these waves is typically 1 for both sine and cosine functions, but transformations can alter this. One common transformation is a vertical shift, which occurs when a constant \( k \) is added to the function. If \( k \) is positive, the graph shifts upward; if negative, it shifts downward.

For example, to graph the function \( y = \sin(x) + 1 \), you would first graph \( y = \sin(x) \) and then shift the entire graph up by 1 unit. This means the original peaks at 1 become peaks at 2, and the valleys at -1 become valleys at 0. The resulting graph will maintain the wave pattern but will be elevated by one unit.

By mastering these concepts, you can effectively graph sine and cosine functions and understand how transformations affect their shapes and positions on the coordinate plane.

To graph the function \( y = \sin(x) + 3 \), it is helpful to start with the basic sine function, \( y = \sin(x) \). The sine function is characterized by its wave-like pattern, starting at the origin (0,0), reaching a peak at \( \frac{\pi}{2} \) (1), crossing the x-axis at \( \pi \) (0), reaching a valley at \( \frac{3\pi}{2} \) (-1), and continuing this oscillation indefinitely in both directions.

The key features of the sine graph include:

• Amplitude: The maximum value is 1 and the minimum value is -1.
• Period: The function completes one full cycle from \( 0 \) to \( 2\pi \).

Next, to graph \( y = \sin(x) + 3 \), we recognize that the addition of 3 shifts the entire graph vertically upwards by 3 units. This means that every point on the sine graph will be moved up by 3. For example:

• The starting point (0,0) moves to (0,3).
• The peak at \( \frac{\pi}{2} \) (1) moves to \( \left(\frac{\pi}{2}, 4\right) \).
• The x-intercept at \( \pi \) (0) moves to \( \left(\pi, 3\right) \).
• The valley at \( \frac{3\pi}{2} \) (-1) moves to \( \left(\frac{3\pi}{2}, 2\right) \).

Continuing this pattern, the graph will maintain its wave-like behavior, but all points will be elevated by 3 units. Therefore, the new minimum value becomes 2 and the new maximum value becomes 4. The graph will still oscillate between these two values, creating a wave that extends infinitely in both directions.

In summary, the graph of \( y = \sin(x) + 3 \) is a vertically shifted version of the standard sine graph, with all points raised by 3 units, resulting in a wave that oscillates between 2 and 4.

In exploring the transformations of sine and cosine functions, we focus on two key concepts: amplitude and reflection. The amplitude of a function is defined as the distance from the midline of the graph to its peaks or valleys. For sine and cosine functions, which are periodic and oscillate, the amplitude can be determined by the coefficient in front of the function. For instance, if we consider the sine function, y = a \sin(x), the amplitude is given by the absolute value of a. If a = 1, the amplitude is 1 unit, while if a = 2, the amplitude becomes 2 units, resulting in a graph that is vertically stretched.

To visualize this, when we plot the function y = 2 \sin(x), we find that the peaks reach 2 units above the midline and the valleys drop to 2 units below it. This transformation is akin to a vertical stretch, similar to transformations encountered in algebra. Conversely, if the amplitude is negative, such as in y = -\sin(x), the graph reflects over the x-axis, flipping the sine wave upside down.

For a more complex example, consider the function y = -\frac{3}{2} \cos(x). Here, the amplitude is \frac{3}{2}, and the negative sign indicates a reflection over the x-axis. The cosine function typically starts at its maximum value, but due to the negative coefficient, it begins at a minimum value instead. As we plot this function, we observe that it descends to its lowest point, rises to a peak, and continues this oscillation, maintaining the amplitude of 1.5 units.

Understanding these transformations allows for a deeper comprehension of how sine and cosine functions behave under various conditions. By manipulating the amplitude and applying reflections, we can predict the shape and position of the graphs effectively.

To graph the function \( y = 2 \sin(x) - 1 \), it is beneficial to first understand the basic sine function, \( y = 2 \sin(x) \), before applying transformations. The sine function typically oscillates between -1 and 1, but with the coefficient of 2, the amplitude is modified. This means the peaks of the graph will reach 2 and the valleys will dip to -2.

Starting from the origin, the graph of \( y = 2 \sin(x) \) will peak at \( \frac{\pi}{2} \) (output of 2), return to 0 at \( \pi \), reach a valley at \( \frac{3\pi}{2} \) (output of -2), and continue this wave pattern indefinitely in both directions. Extending the graph to the left, the valley occurs at \( -\frac{\pi}{2} \) and peaks at \( -\frac{3\pi}{2} \).

Next, to incorporate the vertical shift introduced by the \(-1\) in the function \( y = 2 \sin(x) - 1 \), we shift the entire graph down by 1 unit. This adjustment affects all key points: the peaks now reach an output of 1 (down from 2), and the valleys drop to -3 (down from -2). The centerline of the graph, which was originally at 0, is now at -1.

Thus, the new graph starts at \( -1 \) when \( x = 0 \), peaks at \( \frac{\pi}{2} \) with an output of 1, returns to \( -1 \) at \( \pi \), and reaches a valley at \( \frac{3\pi}{2} \) with an output of -3. Similarly, extending to the left, the valley occurs at \( -\frac{\pi}{2} \), the center returns to \( -1 \) at \( -\pi \), and peaks at \( -\frac{3\pi}{2} \).

In summary, the graph of \( y = 2 \sin(x) - 1 \) oscillates between -3 and 1, with a centerline at -1, demonstrating the effects of amplitude change and vertical shifts on the sine function.

In the study of trigonometric functions, understanding the concepts of amplitude and period is essential for analyzing their graphs. While amplitude indicates the height of the graph, the period describes the width, specifically the distance along the x-axis required to complete one full cycle of the wave. For the sine function, the standard period is 2\pi, which can be determined by observing the graph's behavior from the origin to the point where the wave completes its cycle.

When examining transformations of the sine function, such as \sin(2x), the period changes based on the coefficient of x, denoted as b. In this case, the period can be calculated using the formula:

P = \frac{2\pi}{b}

For \sin(x), since b is 1, the period remains 2\pi. However, for \sin(2x), where b equals 2, the period becomes:

P = \frac{2\pi}{2} = \pi

This indicates that the graph of \sin(2x) completes its cycle more quickly, resulting in a horizontally shrunk appearance compared to the standard sine graph.

Furthermore, if b is greater than 1, the graph shrinks horizontally, while if b is between 0 and 1, the graph stretches horizontally. This relationship allows for straightforward calculations of the period for various sine and cosine functions.

For example, to find the period of \sin\left(\frac{1}{2}x\right), we identify b as \frac{1}{2}. Applying the period formula yields:

P = \frac{2\pi}{\frac{1}{2}} = 4\pi

In another case, for \cos(4\pi x), b is 4\pi, leading to:

P = \frac{2\pi}{4\pi} = \frac{1}{2}

These calculations illustrate how the period of sine and cosine functions can be determined and how the presence of a coefficient in front of x affects the graph's width. Mastering these concepts is crucial for further exploration of trigonometric functions and their applications.