Evaluate, if possible, the given trigonometric function at the indicated value.
$\sin t , t = \frac { 4 \pi } { 3 }$
A) $- \frac { \sqrt { 3 } } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $- \frac { 2 \sqrt { 3 } } { 3 }$
D) $\frac { \sqrt { 3 } } { 2 }$
E) not possible

Question

Quizplus · Accepted Answer

Using the unit circle, we can see that at $t=\frac{4\pi}{3}$ (which is equivalent to $240^\circ$), the terminal side of the angle will lie in the third quadrant where both the $x$- and $y$-coordinates are negative. Therefore, $\sin t$ will also be negative.

Then, using the $30^\circ$-$60^\circ$-$90^\circ$ triangle in the third quadrant (the one with a hypotenuse length of $2$ and short leg length of $1$), we can see that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, so $\sin\left(\frac{4\pi}{3}ight) = -\frac{\sqrt{3}}{2}$.

Therefore, the best choice is $\boxed{	ext{(A)}}.$

Evaluate, If Possible, the Given Trigonometric Function at the Indicated sin⁡t,t=4π3\sin t , t = \frac { 4 \pi } { 3 }sint,t=34π​

Evaluate, If Possible, the Given Trigonometric Function at the Indicated $\sin t , t = \frac { 4 \pi } { 3 }$