Question

Find the exact value of tangent left-parenthesis negative 210 degrees right-parenthesis.

(1 point)
Responses

negative one-half
Image with alt text: negative one-half

negative StartFraction StartRoot 3 EndRoot Over 3 EndFraction
Image with alt text: negative StartFraction StartRoot 3 EndRoot Over 3 EndFraction

Start Fraction Start Root 3 End Root over 2 End Fraction
Image with alt text: Start Fraction Start Root 3 End Root over 2 End Fraction

negative StartRoot 3 EndRoot

Answer 1

To find the exact value of \(\tan(-210^\circ)\), we can use the periodic properties of the tangent function.

1. **Finding a coterminal angle**: Since tangent has a period of \(180^\circ\), we can find a coterminal angle by adding \(180^\circ\) to \(-210^\circ\):

\[
-210^\circ + 180^\circ = -30^\circ
\]

2. **Finding the reference angle**: Now, consider \(-30^\circ\). The reference angle for \(-30^\circ\) is \(30^\circ\), but since we are in the fourth quadrant (where the angle is negative), the tangent function will be negative.

3. **Calculating the tangent**: The tangent of \(30^\circ\) is:

\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\]

Therefore, since \(-30^\circ\) is in the fourth quadrant:

\[
\tan(-30^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\]

Hence, the exact value of \(\tan(-210^\circ)\) is:

\[
-\frac{\sqrt{3}}{3}
\]

So, the correct response is:

**negative StartFraction StartRoot 3 EndRoot Over 3 EndFraction**