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.
a. sin 210°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressio
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. cos 210°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressio
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. tan 210°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressic
b. the function value is undefined.

Explanation:

Step1: Find the reference angle

The reference angle $\theta'$ for an angle $\theta = 210^{\circ}$ (which is in the third - quadrant, since \(180^{\circ}<210^{\circ}<270^{\circ}\)) is given by \(\theta'=\theta - 180^{\circ}\). So, \(\theta'=210^{\circ}-180^{\circ}=30^{\circ}\)

Step2: Determine the sign of the trigonometric functions

In the third - quadrant, \(\sin\theta<0\), \(\cos\theta<0\), and \(\tan\theta>0\)

Step3: Calculate \(\sin210^{\circ}\)

We know that \(\sin\theta=-\sin\theta'\) (because \(\sin\) is negative in the third - quadrant). Since \(\sin30^{\circ}=\frac{1}{2}\), then \(\sin210^{\circ}=-\frac{1}{2}\)

Step4: Calculate \(\cos210^{\circ}\)

We know that \(\cos\theta=-\cos\theta'\) (because \(\cos\) is negative in the third - quadrant). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), then \(\cos210^{\circ}=-\frac{\sqrt{3}}{2}\)

Step5: Calculate \(\tan210^{\circ}\)

We know that \(\tan\theta=\tan\theta'\) (because \(\tan=\frac{\sin}{\cos}\), and both \(\sin\) and \(\cos\) are negative in the third - quadrant, so their ratio is positive). Since \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\), then \(\tan210^{\circ}=\frac{\sqrt{3}}{3}\)

Answer:

For \(\sin210^{\circ}\): A. \(\sin210^{\circ}=-\frac{1}{2}\)
For \(\cos210^{\circ}\): A. \(\cos210^{\circ}=-\frac{\sqrt{3}}{2}\)
For \(\tan210^{\circ}\): A. \(\tan210^{\circ}=\frac{\sqrt{3}}{3}\)