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

The sine and cosine functions create wave-like graphs characterized by their amplitude and period. The amplitude indicates the height of the peaks, while the period defines the width of one complete cycle. The period can be calculated using the equation: $P=\frac{2\pi }{b}$. Transformations such as vertical shifts, amplitude changes, and reflections over the x-axis affect the graph's appearance, enhancing understanding of trigonometric functions.

Graph of Sine and Cosine Function

Understanding the sine and cosine functions is essential for graphing these trigonometric functions effectively. The sine function corresponds to the y-coordinates 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, y = \sin(x), you can start by plotting key points based on the unit circle. For instance, at x = 0, y = \sin(0) = 0; at x = \frac{\pi}{2}, y = \sin\left(\frac{\pi}{2}\right) = 1; at x = \pi, y = \sin(\pi) = 0; at x = \frac{3\pi}{2}, y = \sin\left(\frac{3\pi}{2}\right) = -1; and at x = 2\pi, y = \sin(2\pi) = 0. Connecting these points reveals a smooth wave pattern that repeats every 2\pi.

Similarly, for the cosine function, y = \cos(x), the key points are: at x = 0, y = \cos(0) = 1; at x = \frac{\pi}{2}, y = \cos\left(\frac{\pi}{2}\right) = 0; at x = \pi, y = \cos(\pi) = -1; at x = \frac{3\pi}{2}, y = \cos\left(\frac{3\pi}{2}\right) = 0; and at x = 2\pi, y = \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 starts at 0.

In wave terminology, the highest points of the graph are referred to as crests or peaks, while the lowest points are called troughs or valleys. The sine and cosine functions typically oscillate between 1 and -1, but transformations can alter these ranges. One common transformation is the 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 1 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.

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

• A.
• B.
• C.
• D.

Answer: (B)

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

• A. $y=50$
• B. $y=51$
• C. $y=49$
• D. $y=50+\frac{\sqrt3}{2}$

Answer: (C)

Example 1

To graph the function \( y = \sin(x) + 3 \), it's beneficial to start with the basic sine function, \( y = \sin(x) \), which is a periodic wave oscillating between -1 and 1. The sine function begins at the origin (0,0), reaches its maximum at \( \frac{\pi}{2} \) (1), crosses the x-axis at \( \pi \) (0), reaches its minimum at \( \frac{3\pi}{2} \) (-1), and continues this pattern indefinitely in both directions.

When we introduce the transformation \( +3 \) in the function \( y = \sin(x) + 3 \), it results in a vertical shift of the entire sine wave upwards by 3 units. This means that every point on the original sine graph will be moved up by 3. Consequently, the new key points will be:

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

Similarly, for negative values of \( x \), the points will also shift up by 3 units. For example, the point at \( -\frac{\pi}{2} \) will now be at \( (-\frac{\pi}{2}, 2) \), and the point at \( -\pi \) will be at \( (-\pi, 3) \).

By connecting these transformed points, the graph of \( y = \sin(x) + 3 \) will exhibit the same wave-like behavior as the original sine function, but it will oscillate between 2 and 4 instead of -1 and 1. This results in a complete wave pattern that extends infinitely in both directions, maintaining the periodic nature of the sine function while being vertically shifted.

Amplitude and Reflection of Sine and Cosine

In this lesson, we explore the transformations of the sine and cosine functions, focusing on amplitude and reflection. Understanding these transformations is essential as they are similar to those encountered in algebra, making them more intuitive.

The amplitude of a sine or cosine function is defined as the distance from the midline of the graph to its peaks or valleys. For instance, if the sine function is represented as \(y = \sin(x)\), the amplitude is 1, as the distance from the midline to the peak is one unit. If we modify the function to \(y = 2\sin(x)\), the amplitude becomes 2, indicating that the peaks are now twice as high. This transformation can be visualized as a vertical stretch of the graph.

To calculate the amplitude, you can identify the coefficient in front of the sine or cosine function. A positive coefficient results in a standard wave, while a negative coefficient reflects the graph over the x-axis. For example, the function \(y = -\sin(x)\) flips the sine wave upside down, demonstrating how a negative amplitude affects the graph.

When combining both amplitude changes and reflections, such as in the function \(y = -\frac{3}{2}\cos(x)\), the amplitude is \(1.5\) (or \(\frac{3}{2}\)). The negative sign indicates that the graph will be reflected over the x-axis. Starting from a low point, the cosine function will rise to a peak, then descend, creating a wave pattern that is inverted compared to the standard cosine graph.

In summary, the amplitude and reflection of sine and cosine functions are crucial for understanding their behavior. By adjusting the coefficients, you can manipulate the height and orientation of these periodic functions, allowing for a deeper comprehension of their graphical representations.

Determine the value of $y=-2\cdot\sin\left(-\frac{3\pi}{2}\right)+10$ without using a calculator or the unit circle.

• A. $y=8$
• B. $y=10$
• C. $y=-2$
• D. $y=12$

Answer: (A)

Graph the function $y=-3\cdot\cos\left(x\right)$.

• A.
• B.
• C.
• D.

Answer: (B)

Example 1

To graph the function \( y = 2 \sin(x) - 1 \), it is beneficial to first understand the basic sine function, \( y = \sin(x) \), and then apply transformations to it. The sine function has a standard amplitude of 1, which means it oscillates between -1 and 1. However, in this case, the function is modified by a coefficient of 2, which affects the amplitude.

The amplitude of the sine function is determined by the coefficient in front of the sine term. For \( y = 2 \sin(x) \), the amplitude is 2, meaning the graph will oscillate between -2 and 2. The peaks occur at \( \frac{\pi}{2} \) and the valleys at \( \frac{3\pi}{2} \). The graph starts at the origin (0,0), reaches its maximum at \( \left(\frac{\pi}{2}, 2\right) \), returns to zero at \( (\pi, 0) \), reaches its minimum at \( \left(\frac{3\pi}{2}, -2\right) \), and continues this wave pattern indefinitely in both directions.

Next, we incorporate the vertical shift introduced by the -1 in the function \( y = 2 \sin(x) - 1 \). This shift moves the entire graph down by 1 unit. Consequently, the new peaks will now be at \( 1 \) (from the original peak of 2) and the valleys will be at \( -3 \) (from the original valley of -2). The centerline of the graph, which was originally at \( y = 0 \), is now at \( y = -1 \).

To summarize the transformations: the graph starts at \( (-1) \) instead of \( 0 \), reaches a peak at \( \left(\frac{\pi}{2}, 1\right) \), returns to the centerline at \( (\pi, -1) \), and reaches a valley at \( \left(\frac{3\pi}{2}, -3\right) \). This pattern continues as the graph extends to the left and right, maintaining the same wave-like behavior but shifted downwards.

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

Period of Sine and Cosine Functions

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 wave, the period describes the width of the graph, 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 means it takes this distance to return to its starting point after one complete oscillation.

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

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

This indicates that the graph of \sin(2x) completes its cycle in a shorter distance, specifically at \pi, compared to the standard sine function.

Furthermore, if the value of b is greater than 1, the graph horizontally shrinks, resulting in a shorter period. Conversely, if b is between 0 and 1, the graph stretches horizontally, leading to a longer period. This relationship allows for straightforward calculations of the period based on the coefficient in front of x.

To illustrate this, consider two examples. For the function \sin\left(\frac{1}{2}x\right), the value of b is \frac{1}{2}, leading to a period of:

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

In another example, for \cos(4\pi x), where b is 4\pi, the period is calculated as:

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

These calculations demonstrate how the period of sine and cosine functions can be determined without graphing, providing a clear understanding of how the transformations affect the wave patterns. Mastery of these concepts is crucial for further studies in trigonometry and its applications.

Given below is the graph of the function $y=\sin\left(bx\right)$. Determine the correct value for b.

• A. $b=\pi$
• B. $b=2$
• C. $b=\frac12$
• D. $b=4$

Answer: (D)

The Period for the function $y=\cos\left(bx\right)$ is $T=20\pi$. Determine the correct value of b.

• A. $b=\frac{1}{10}$
• B. $b=10$
• C. $b=20$
• D. $b=\frac{1}{20}$

Answer: (A)

Here’s what students ask on this topic:

What do the graphs of sine and cosine functions look like?

The graphs of sine and cosine functions are wave-like and periodic. The sine function, y = sin(x), starts at 0, reaches a maximum of 1 at π/2, returns to 0 at π, reaches a minimum of -1 at 3π/2, and returns to 0 at 2π, forming a smooth wave. The cosine function, y = cos(x), starts at 1, drops to 0 at π/2, reaches -1 at π, returns to 0 at 3π/2, and back to 1 at 2π, also forming a wave. Both functions repeat this pattern indefinitely.

How do you determine the amplitude of sine and cosine functions?

The amplitude of sine and cosine functions is the distance from the midline to the peak or trough of the wave. It is determined by the coefficient in front of the sine or cosine function. For example, in y = A sin(x) or y = A cos(x), the amplitude is |A|. If A is positive, the graph maintains its original orientation; if A is negative, the graph is reflected over the x-axis.

What is the period of sine and cosine functions, and how is it calculated?

The period of sine and cosine functions is the distance along the x-axis required to complete one full cycle of the wave. It is calculated using the formula P = 2π/b, where b is the coefficient of x inside the function. For example, in y = sin(bx) or y = cos(bx), if b = 1, the period is 2π. If b = 2, the period is π, indicating the wave completes a cycle twice as fast.

How do vertical shifts affect the graphs of sine and cosine functions?

Vertical shifts in sine and cosine functions occur when a constant k is added or subtracted from the function. For example, in y = sin(x) + k or y = cos(x) + k, the entire graph shifts up by k units if k is positive, and down by k units if k is negative. This shift does not affect the amplitude or period but changes the midline of the wave to y = k.

What happens to the sine and cosine graphs when the amplitude is negative?

When the amplitude of sine and cosine functions is negative, the graph is reflected over the x-axis. For example, in y = -A sin(x) or y = -A cos(x), the wave is flipped upside down. The peaks become troughs and vice versa, but the absolute value of the amplitude remains the same, indicating the height of the wave from the midline.