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Chapter 5: Problem 74

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

Short Answer

Expert verified
So, the simplified form of \( \tan (\pi + \theta) \) is \( \tan (\theta) \).

Step by step solution

01

Identify the pattern of tangent

First, remember that tangent has a period of \( \pi \). So \( \tan(\theta) \) is same for all \( \theta + n\pi \) where n is any integer. So, \( \tan (\pi+\theta) \) equals to \( \tan (\theta) \).
02

Mapping on unit circle

The hint to why the results are equal is in the unit circle. When a point makes an angle \( \theta \) with the base in a unit circle and if we add \( \pi \) to that angle, we get the same tangent value but in the opposite quadrant of the circle. And from the properties of tangent, we know that \( \tan (\pi + \theta) = \tan (\theta) \).
03

Confirmation with graph

`To confirm this, plot the graph of \( y=\tan(x) \) and \( y=\tan(\pi+x) \) in a graphing utility. You will notice that the two functions produce identical waves, confirming that they are indeed the same.

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