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Chapter 5: Problem 72

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\cos (\pi+x)$$

Short Answer

Expert verified
\(\cos(\pi + x) = -\cos(x)\). This simplification can be visually confirmed by comparing the graphs of y=\(\cos(x)\) and y=-\(\cos(x)\), where the latter is found to be a reflected version of the former about the x-axis.

Step by step solution

01

Use Cosine Properties

By knowing the property of cosine, we have that: \[\cos(\pi + x) = -\cos(x)\]. The cosine function has a property that \(\cos(\pi + x) = -\cos(x)\). This is due to the fact that \(\cos(\pi + x)\) means that x is shifted by \(\pi\) in the graph which causes the cosine graph to shift, and the value becomes negative.
02

Draw the Graph Using Graphing Utility

Graph the functions y=\(\cos(x)\) and y=-\(\cos(x)\) with the same graphing utility. The graph y=\(\cos(x)\) is a wave oscillating between -1 and 1, starting at 1. When \(\pi\) added to x, it becomes y=-\(\cos(x)\) which flips the graph upside down.
03

Confirm Graphically

By comparing the two graphs, it can be seen that y=-\(\cos(x)\) is the reflection of y=\(\cos(x)\) about the x-axis, confirming our initial calculation.

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