3. which one doesn’t belong? give a convincing argument for each answer choice. sin(2π/3) cos(5π/6)…

Explanation:

Step1: Calculate $\sin(\frac{2\pi}{3})$

We know that $\sin(\frac{2\pi}{3})=\sin(120^{\circ})=\frac{\sqrt{3}}{2}$ using the unit - circle or angle - addition formulas.

Step2: Calculate $\cos(\frac{5\pi}{6})$

$\cos(\frac{5\pi}{6})=\cos(150^{\circ})=-\frac{\sqrt{3}}{2}$ as $\cos(150^{\circ})=\cos(180^{\circ} - 30^{\circ})=-\cos(30^{\circ})$.

Step3: Calculate $\sin(240^{\circ})$

$\sin(240^{\circ})=\sin(180^{\circ}+60^{\circ})=-\sin(60^{\circ})=-\frac{\sqrt{3}}{2}$ using the angle - addition formula $\sin(A + B)=\sin A\cos B+\cos A\sin B$ with $A = 180^{\circ}$ and $B = 60^{\circ}$.

Step4: Calculate $\sin(\frac{11\pi}{6})$

$\sin(\frac{11\pi}{6})=\sin(330^{\circ})=\sin(360^{\circ}-30^{\circ})=-\sin(30^{\circ})=-\frac{1}{2}$ using the angle - addition formula $\sin(A - B)=\sin A\cos B-\cos A\sin B$ with $A = 360^{\circ}$ and $B = 30^{\circ}$.

Argument for $\sin(\frac{2\pi}{3})$ not belonging:

It is the only positive - valued expression among the four. The other three expressions $\cos(\frac{5\pi}{6})$, $\sin(240^{\circ})$, and $\sin(\frac{11\pi}{6})$ are all negative.

Argument for $\cos(\frac{5\pi}{6})$ not belonging:

It is the only cosine function among the four expressions. The other three are sine functions.

Argument for $\sin(240^{\circ})$ not belonging:

It is the only angle given in degrees while the other three are given in radians.

Argument for $\sin(\frac{11\pi}{6})$ not belonging:

Its value is $-\frac{1}{2}$, while the other three values have a magnitude of $\frac{\sqrt{3}}{2}$.

There is no single "correct" answer as it depends on the chosen criterion.