Question

the given unit circle has been divided into eight equal arcs, corresponding to t - values of (0, \frac{pi}{4}, \frac{pi}{2}, \frac{3pi}{4}, pi, \frac{5pi}{4}, \frac{3pi}{2}, \frac{7pi}{4}), and (2pi). complete parts (a) and (b).
a. use the ((x,y)) coordinates in the figure to find the value of (sin 2pi). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (sin 2pi = 0) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the solution is undefined.
b. use periodic properties and the answer from part (a) to find the value of (sin 14pi). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (sin 14pi=) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the solution is undefined.

Explanation:

Step1: Identify point for $2\pi$

On unit circle, $t=2\pi$ corresponds to $(1,0)$. For $\sin t$, we take the $y$-coordinate.
$\sin 2\pi = 0$

Step2: Use sine periodicity

Sine has period $2\pi$, so $\sin(t + 2k\pi) = \sin t$ for integer $k$. Rewrite $14\pi$:
$14\pi = 2\pi + 6\times2\pi$
$\sin 14\pi = \sin(2\pi + 6\times2\pi) = \sin 2\pi = 0$

Answer:

a. A. $\sin 2\pi = 0$
b. A. $\sin 14\pi = 0$