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

Question

(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
Image with alt text: negative StartRoot 3 EndRoot

Answer 1

To find the exact value of $\tan (-{210}^{\circ })$, we can use the periodicity of the tangent function and the fact that $\tan (-\theta )=-\tan (\theta )$.

First, let's convert $-{210}^{\circ }$ to a positive angle by adding ${360}^{\circ }$ (the full rotation):

$$-{210}^{\circ }+{360}^{\circ }={150}^{\circ }$$

Now we can find:

$$\tan (-{210}^{\circ })=\tan ({150}^{\circ })$$

Next, we know that ${150}^{\circ }$ is in the second quadrant, where the tangent function is negative. We can find $\tan ({150}^{\circ })$ by using its reference angle, which is ${180}^{\circ }-{150}^{\circ }={30}^{\circ }$:

$$\tan ({150}^{\circ })=-\tan ({30}^{\circ })$$

The value of $\tan ({30}^{\circ })$ is $\frac{1}{\sqrt{3}}$, thus:

$$\tan ({150}^{\circ })=-\frac{1}{\sqrt{3}}$$

Finally, putting this together, we find:

$$\tan (-{210}^{\circ })=-\tan ({150}^{\circ })=-(-\frac{1}{\sqrt{3}})=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$$

So, the exact value of $\tan (-{210}^{\circ })$ is:

$$\boxed{{\displaystyle -\frac{\sqrt{3}}{3}}}$$