Use the definitions (not a calculator) to evaluate the six trigonometric functions of the given angle: $2 \pi$
A) $\sin 2 \pi = 0 , \cos 2 \pi = 1 , \tan 2 \pi = 0 , \sec 2 \pi = 1 , \csc 2 \pi \text { is undefined, } \cot 2 \pi \text { is undefined. }$
B) $\sin 2 \pi = 1 , \cos 2 \pi = 1 , \tan 2 \pi = 1 , \sec 2 \pi = 1 , \csc 2 \pi = 1 , \cot 2 \pi = 1$
C) $\sin 2 \pi = 0 , \cos 2 \pi = 1 , \tan 2 \pi \text { is undefined, } \sec 2 \pi \text { is undefined. } \csc 2 \pi = 0 , \cot 2 \pi = 1$
D) $\sin 2 \pi = - 1 \cdot \cos 2 \pi = 1 \cdot \tan 2 \pi = - 1 \cdot \sec 2 \pi = 1 \cdot \csc 2 \pi = 1 \cdot \cot 2 \pi = 1$
E) $\sin 2 \pi = 1 , \cos 2 \pi = 0 , \tan 2 \pi \text { is undefined, } \sec 2 \pi \text { is undefined. } \csc 2 \pi = - 1 , \cot 2 \pi = 0$

Question

Quizplus · Accepted Answer

Using the unit circle, we see that an angle of $2\pi$ radians corresponds to a full revolution and ends up on the point $(1,0)$. Therefore, $\sin 2\pi = 0$ and $\cos 2\pi = 1$.

Since $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$, we have $	an 2\pi = \frac{0}{1} = 0$. Also, $\csc 	heta = \frac{1}{\sin 	heta}$ and $\sec 	heta = \frac{1}{\cos	heta}$, so $\csc 2\pi$ and $\sec 2\pi$ are undefined.

Finally, $\cot 	heta = \frac{1}{	an 	heta} = \frac{\cos 	heta}{\sin 	heta}$. Since $\sin 2\pi = 0$, this means $\cot 2\pi$ is undefined. Therefore, the correct choice is $\boxed{	extbf{(A)}}$.

Use the Definitions (Not a Calculator) to Evaluate the Six 2π2 \pi2π

Use the Definitions (Not a Calculator) to Evaluate the Six $2 \pi$