Respuesta :
Given the expression below:
[tex]\sin (\frac{2\pi}{3})[/tex]To find the exact value of the expression, let us determine the quadrant of the expression. It should be noted that the value of angles compare with the quadrants is as shown below
[tex]\begin{gathered} First\text{ quadrant, the measure of reference angle in radian is } \\ 0-\frac{\pi}{2} \end{gathered}[/tex][tex]\begin{gathered} \text{second quadrant, the measure of reference angle in radian is} \\ \frac{\pi}{2}-\pi \end{gathered}[/tex][tex]\begin{gathered} \text{third quadrant, the measure of reference angle in radian is} \\ \pi-\frac{3\pi}{2} \end{gathered}[/tex][tex]\begin{gathered} \text{fourth quadrant, the measure of reference angle in radian is} \\ \frac{3\pi}{2}-2\pi \end{gathered}[/tex]It can be observed that the expression given in the question is a fraction of (pi), greater than half of (pi) but less than (pi). This means that it lies in the second quadrant.
It should be noted that sine is positive in the second quadrant
The equivalent of the expression in the first quadrant is as shown below:
[tex]\begin{gathered} \sin (\frac{2\pi}{3})=\sin (\pi-\frac{2\pi}{3}) \\ =\sin (\frac{3\pi-2\pi}{3}) \\ =\sin (\frac{\pi}{3}) \end{gathered}[/tex][tex]\begin{gathered} \text{Therefore,} \\ \sin (\frac{2\pi}{3}),in\text{ second quadrant is the same } \\ \sin (\frac{\pi}{3}),in\text{ first quadrant.} \\ \sin (\frac{\pi}{3})=\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]Hence, the exact value of the expression is √3/2
