Problem 39 - Algebra 2

Question

Graph one cycle of \(y=\cos \theta,\) one cycle of \(y=-\cos \theta,\) and one cycle of \(y=\cos (-\theta)\) on the same set of axes. Use the unit circle to explain any relationships you see among these graphs.

Step-by-Step Solution

Answer

The graph of \(y = \cos \theta\) is a cosine wave starting from (1,0). The graph of \(y = -\cos \theta\) is an inverted cosine wave as it is a reflection about the x-axis. The graph of \(y = \cos(-\theta)\) is identical to \(y = \cos \theta\) as cosine is an even function.

Step 1: Graph of \(y = \cos \theta\)

Starting with the graph of \(y = \cos \theta\). On the unit circle, the horizontal axis represents the value of cosine while the angle is shown by theta. A complete cycle of cosine starts at \((1,0)\) for \(\theta=0\) and moves along the unit circle reaching start point again at \(\theta=2\pi\). The graph is wave-like with peaks at 1 (which occur at \(2\pi n\) where n is an integer) and troughs at -1 (which occur at \((2n+1)\pi\)).

Step 2: Graph of \(y = -\cos \theta\)

The graph of \(y = -\cos \theta\) is basically a reflection of the graph \(y = cos \theta\) about the x-axis because it involves negating the value of the original function. This means that the all the peaks and troughs are now inverted.

Step 3: Graph of \(y = \cos (-\theta)\)

For the function \(y = \cos(-\theta)\), the angle \(\theta\) is negated. However, cosine is an even function meaning that it is unchanged when the input is negated i.e, \(\cos \theta = \cos (-\theta)\). So, the graph will be identical to the graph of \(y = \cos \theta\).

Step 4: Observation from the same set of axes

From the same set of axes for all these trigonometric functions, it is observable that the relationships between these functions can be described based on the symmetries in the unit circle. The function \(y=-\cos \theta\) is a reflection of \(y=\cos \theta\) about the x-axis. Also, there is no difference between the functions \(y=\cos \theta\) and \(y=\cos (-\theta)\) due to the property of cosine being an even function.

Key Concepts

Unit Circle

The unit circle is a crucial tool in understanding trigonometric functions like the cosine function. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This simple circle allows us to associate angles with coordinates. As you move around the circle, your current position can be represented by an angle \(\theta\), which correlates to a point \((x, y)\). In our case, the \(x\)-coordinate represents the cosine of the angle: \(x = \cos\theta\).

Starting from the point \((1, 0)\), which represents \(\theta = 0\), a complete cycle around the unit circle (back to the starting position) occurs at \(\theta = 2\pi\). The value of cosine smoothly transitions from 1 to -1 and back, defining the wave-like characteristic of the cosine function when graphed. Using the unit circle connects the geometric meaning with the algebraic function, providing a visual way to comprehend how \(\cos\theta\) behaves over its cycle.

Cosine Function

The cosine function is a periodic function, meaning it repeats its values at regular intervals. For cosine, this interval or period is \(2\pi\). This periodicity is what creates the familiar smooth, wave-like pattern of its graph.

In a single period starting at \(\theta = 0\):

• The graph begins at the highest point (1)
• Decreases to zero at \(\theta = \frac{\pi}{2}\)
• Reaches the lowest point (-1) at \(\theta = \pi\)
• Returns to zero at \(\theta = \frac{3\pi}{2}\)
• Completes the cycle at the starting height (1) at \(\theta = 2\pi\)

The cosine function's symmetry and periodicity make it predictable and easy to work with, characteristics leveraged in graphing and transformations.

Even and Odd Functions

Understanding even and odd functions helps simplify the analysis and graphing of functions like cosine and others. An even function has the property \(f(-x) = f(x)\) for all \(x\). This means that the graph of an even function is symmetrical about the y-axis.

The cosine function is an example of an even function. For it,

• \(\cos(\theta) = \cos(-\theta)\)

Thus, the cosine function doesn't change when you negate its input. This property explains why the graph of \(y = \cos(-\theta)\) is exactly the same as that of \(y = \cos \theta\). Recognizing a function as even can simplify calculations and graphing tasks.

Graph Reflections

Graph reflections involve flipping a graph across an axis, conveying how the graph's shape changes visually. When graphing, reflection about the x-axis is commonly encountered.

To reflect a graph over the x-axis, you simply multiply the function by -1. For example, the transformation from \(y = \cos \theta\) to \(y = -\cos \theta\) indicates reflection:

• Peaks above the axis in \(\cos \theta\) will become troughs below the axis in \(-\cos \theta\) and vice versa.

This reflection is consistent with our intuitive understanding of negatives; every "+" becomes a "-". The knowledge of reflections allows one to predict the behavior of function transformations easily when creating and analyzing graphs.