Question

Math

Posted 3 months ago

```
On the following unit circle, $\theta$ is in radians and $\tan (\theta)=\frac{-0.99}{0.1}=-9.9$.
Without a calculator, evaluate the following expressions to the nearest hundredth or as an exact ratio.
\begin{tabular}{ll}
Expression & Value \\
\hline $\tan (\pi+\theta)$ & $\square$ \\
$\tan (2 \pi+\theta)$ & $\square$
\end{tabular}
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Recall the periodic properties of the tangent function, specifically that $\tan(\theta)$ is periodic with period $\pi$. This means that $\tan(\theta + \pi) = \tan(\theta)$

step 2

Apply the periodic property to $\tan(\pi + \theta)$. Since $\tan(\theta) = -9.9$, we have $\tan(\pi + \theta) = \tan(\theta) = -9.9$

step 3

Similarly, apply the periodic property to $\tan(2\pi + \theta)$. Since $\tan(\theta)$ is periodic with period $\pi$, $\tan(2\pi + \theta) = \tan(\theta) = -9.9$

[question 1] Answer

$\tan(\pi + \theta) = -9.9$

$\tan(2\pi + \theta) = -9.9$

Key Concept

Periodicity of the Tangent Function

Explanation

The tangent function has a period of $\pi$, which means that $\tan(\theta + n\pi) = \tan(\theta)$ for any integer $n$. This property allows us to find the value of the tangent function for angles that differ by an integer multiple of $\pi$.

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