I know what the graph of:
y=cos^-1(cos(x))
looks like.
However, can someone explain to me how you would know how to sketch it without simply subbing in values to get points. Something about restrictions and how it is a periodic function possibly?
To sketch the graph of y = cos^(-1)(cos(x)), it is helpful to understand the properties of the inverse cosine function and the periodic nature of the cosine function.
1. Understanding inverse cosine function (arccosine):
The inverse cosine function, denoted as arccos or cos^(-1), returns the angle whose cosine is a given value. The domain of arccos is -1 ≤ x ≤ 1, and the range is 0 ≤ y ≤ π.
2. Periodicity of the cosine function:
The cosine function has a period of 2π, meaning it repeats its values every 2π units. Therefore, cos(x) = cos(x + 2πn), where n is an integer.
Now, let's sketch the graph step by step:
Step 1: Identify the domain and range restrictions:
Since y = cos^(-1)(cos(x)), the domain of this function is the same as the range of cos(x), which is -1 ≤ x ≤ 1. However, to properly sketch the graph, we need to consider multiples of 2π due to the periodicity of the cosine function.
Step 2: Determine key points within the domain:
To find key points on the graph, we need to consider values of x that give specific values for cos(x). Instead of randomly selecting values, we can use the periodic nature of cos(x) to determine the key points within the given domain.
For arccos(cos(x)), the output will always be within the range of 0 ≤ y ≤ π. Therefore, our key points for x can be chosen from the range -1 ≤ x ≤ 1.
Let's consider a few values of x within this range and find their corresponding y values:
When x = -1, cos(x) = cos(-1) ≈ 0.54, so y = arccos(0.54) ≈ 0.99π.
When x = 0, cos(x) = cos(0) = 1, so y = arccos(1) = 0.
When x = 1, cos(x) = cos(1) ≈ 0.54, so y = arccos(0.54) ≈ 0.99π.
These points (-1, 0.99π), (0, 0), and (1, 0.99π) can be plotted on the graph.
Step 3: Sketching the graph:
Based on the points we obtained and the knowledge that the function is periodic, we can sketch the graph. Here's what it would look like:
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-3π/2 -π -π/2 0 π/2 π 3π/2 2π
The graph consists of line segments connecting the points (-1, 0.99π), (0, 0), and (1, 0.99π). It extends infinitely in both the positive and negative x-directions, following the periodicity of the cosine function.
Please note that the scale, proportions, and precise shape may not be accurately conveyed in this text-based format, but it should give you a general idea of the graph's appearance.
In summary, by understanding the properties of the inverse cosine function and the periodic nature of the cosine function, you can determine the graph of y = cos^(-1)(cos(x)) without solely relying on substituting values.