<s>[INST] Find the equivalent value of radians A function $f$ is called periodic with period $T$ if $f (x + T ) = f (x)$ (for all x) and $T$ is the smallest positive number with this property. The sine and cosine functions are periodic with period $T = 2\pi$ because the radian measures $x$ and $x + 2\pi k$ correspond to the same point on the unit circle for any integer $k$:
$$\textbf{sin x = sin(x+2pi k)}$$
Now that 
The Point $P$ on the unit circle corresponding to the angle $\theta=\displaystyle\frac{4\pi}3$ lies opposite with angle $\theta=\displaystyle\frac{\pi}3$.
$$\sin\frac{4\pi}3 = - \sin\frac{\pi}3$$
But how do I find the relation as to why $\displaystyle\sin\frac{4\pi}3 = - \sin\frac{\pi}3$ ?
I tried 
$$\pi/3 = \pi/3+ 2\pi (1/2)$$
According to the definition above so that $\pi/3 = 4\pi/3$. But with this, it'll take a while to find the $k$ is $1/2$. Are there any short ways to find out why  $\sin\frac{4\pi}3 = - \sin\frac{\pi}3$? Assuming you never know that $2\pi =  360^{\circ} $.
Additional information I read:
In general, two radian measures represent the same angle if the corresponding rotations differ by an integer multiple of $2\pi$.
But I couldn't seem to link $\sin\frac{4\pi}3 = - \sin\frac{\pi}3$ with this approach.
 [/INST] To see why $\sin\left(\dfrac{4\pi}{3}\right) = \sin\left(-\dfrac{\pi}{3}\right)$, it helps to consider symmetry.

Method 1:  Since the point $(\cos(\pi + \theta, \sin(\pi + \theta))$ is the reflection of the point $(\cos\theta, \sin\theta)$ with respect to the origin, 
$$\sin(\pi + \theta) = -\sin\theta$$
Hence, 
$$\sin\left(\frac{4\pi}{3}\right) = \sin\left(\pi + \frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)$$
Since the point $(\cos(-\theta), \sin(-\theta))$ is the reflection of the point $(\cos\theta, \sin\theta)$ with respect to the $x$-axis, 
$$\sin(-\theta) = -\sin\theta$$
Hence, 
$$\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)$$
By substitution, we obtain
$$\sin\left(\frac{4\pi}{3}\right) = \sin\left(-\frac{\pi}{3}\right)$$
Method 2:  Since the point $(\cos(\pi - \theta), \sin(\pi - \theta))$ is the reflection of the point $(\cos\theta, \sin\theta)$ with respect to the $y$-axis, 
$$\sin(\pi - \theta) = \sin\theta$$
Since 
$$\frac{4\pi}{3} = \pi - \left(-\frac{\pi}{3}\right)$$
we obtain 
$$\sin\left(\frac{4\pi}{3}\right) = \sin\left[\pi - \left(-\frac{\pi}{3}\right)\right] = \sin\left(-\frac{\pi}{3}\right)$$
On a more abstract level:  
Two angles have the same value of sine if they have the same $y$-coordinate.  Hence, $\sin\theta = \sin\varphi$ if $\theta = \varphi$ or $\theta = \pi - \varphi$.  Since coterminal angles have the same sine, we have the general solution that $\sin\theta = \sin\varphi$ if 
$$\theta = \varphi + 2k\pi, k \in \mathbb{Z}$$
or 
$$\theta = \pi - \varphi + 2m\pi, m \in \mathbb{Z}$$
At the risk of obscuring the symmetry relationships, the two equations can be expressed in the form
$$\theta = (-1)^n\varphi + n\pi, n \in \mathbb{Z}$$
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