Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator. 210° select the correct choice below and fill in any answer boxes within your choice. sin 210° = -1/2 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a b. the function value is undefined. select the correct choice below and fill in any answer boxes within your choice. cos 210° = -√3/2 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a b. the function value is undefined. select the correct choice below and fill in any answer boxes within your choice. tan 210° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a b. the function value is undefined.

Explanation:

Step1: Recall angle - quadrant relationship

The angle $210^{\circ}=180^{\circ} + 30^{\circ}$, so it is in the third - quadrant. In the third - quadrant, sine, cosine and tangent have specific signs. $\sin\theta<0$, $\cos\theta<0$ and $\tan\theta>0$.

Step2: Find the sine value

We know that $\sin(180^{\circ}+\alpha)=-\sin\alpha$. For $\alpha = 30^{\circ}$, $\sin210^{\circ}=\sin(180^{\circ}+30^{\circ})=-\sin30^{\circ}=-\frac{1}{2}$.

Step3: Find the cosine value

We know that $\cos(180^{\circ}+\alpha)=-\cos\alpha$. For $\alpha = 30^{\circ}$, $\cos210^{\circ}=\cos(180^{\circ}+30^{\circ})=-\cos30^{\circ}=-\frac{\sqrt{3}}{2}$.

Step4: Find the tangent value

We know that $\tan\theta=\frac{\sin\theta}{\cos\theta}$. So $\tan210^{\circ}=\frac{\sin210^{\circ}}{\cos210^{\circ}}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$.

Answer:

$\sin210^{\circ}=-\frac{1}{2}$, $\cos210^{\circ}=-\frac{\sqrt{3}}{2}$, $\tan210^{\circ}=\frac{\sqrt{3}}{3}$