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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.

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    Periodicity isn't enough. You need relations such as $\sin (\pi + x) = - \sin x$ and $\sin (-x) = - \sin x$ – 2017-02-05
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    That's not part of the period. k must be an integer and 1/2 doesn't work. It's the symmetric nature of trig that sin (x)=-sin (-x) *and* that sin (x)=sin (pi - x). so sin (4 pi/3)=-sin (-4 pi/3)=-sin (-4 pi/3+ 2 pi) =-(sin 2 pi /3) =-sin (pi - 2 pi/3)=-sin pi/3. – 2017-02-05
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    Hey I actually wanted to said that 1/2 is not valid because not an integer, I missed that, btw hmm... i got no shortest choices rather than to compute it twice rigth? via $4\pi/3$ then $2\pi/3$ then $\pi/3$ – 2017-02-05

2 Answers 2

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To see why $\sin\left(\dfrac{4\pi}{3}\right) = \sin\left(-\dfrac{\pi}{3}\right)$, it helps to consider symmetry.

symmetry_relationships_for_sine_and_cosine

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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    _Thank you! very nice explanation! – 2017-02-05
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According to what you are given, an angle $\theta$ and an angle $\theta + 2 \pi k$ are defined as coterminal, with rays from the origin terminating at the same point on the circle. This means that an angle of $2 \pi$ represents a full turn around the circle, equivalent to 360 degrees.

Therefore to get from an angle to the opposite angle, you add a half turn or $1/2 * 2 \pi = \pi.$ The angle $\theta + \pi$ will give the opposite ray to $\theta.$

A more productive method is to draw a circle with center at the origin with radius $= r$. The radius $r$ can be any positive number measured outward from the origin. (Later in polar graphs, a negative radius is interpreted as measuring backwards, along the opposite ray.)

Then we define (no depth, just what they are) for any point on the circle $P(x,y), sin \theta = y/r, cos \theta = x/r, tan \theta = y/x = $slope of OP. And similarly define angle $\theta = $arc/$r$ where $arc$ = the part of the circle traced by the end of the ray going from an angle of zero to the terminal ray of $\theta$. Set up this way, radian measure is a ratio just like sine, cosine, and tangent. It is clearly defined and visible. It is a pure number with no units (technicaly one should not say "radians" -- a radian is not a unit.) Clearly the full circle or 360 degrees is $2 \pi r / r = 2 \pi,$ the half circle or 180 degrees is $\pi$ and so on.