Finding the value of $\sin {\frac{31 \pi}3}$

Question

The task was to find out the value of $$\sin\frac{31\pi}3$$

This is a example in my book in which following steps are shown: $$\begin{align} \sin\frac{31\pi}3& =\sin\left(10\pi +\frac\pi3\right)\\ &=\sin\frac\pi3\\ &=\frac{\sqrt3}2 \end{align}$$

I cannot understand step 3, $=\sin\frac\pi3$

Answer 1

The sine function is periodic with period $2\pi$. This means that $\sin\theta = \sin(\theta + 2\pi) \ \forall \ \theta$. One can apply this identity five times to get rid of the $10 \pi$ term.

Answer 2

Multiples of $2\pi$ do not matter because $\sin$ is periodic of period $2\pi$.

Answer 3

You can confirm this by using the angle-sum formula:

$$\begin{align} \sin(10\pi + \pi/3) & = \sin(10\pi)\cos(\pi/3) + \cos(10 \pi)\sin(\pi/3) \\ \\ & = 0\cdot\frac 12 + 1 \cdot \sin (\pi/3) \\ \\ & = \sin(\pi/3)\\ \\ & = \sqrt 3/2\end{align}$$

Answer 4

The sine function is periodic with period $2\pi$. A periodic function is a function that repeats its values in regular intervals or periods. A function is said to be periodic with period $P$, where $P$ being a nonzero constant if we have $$ f(x+P) = f(x) $$ for all values of $x$ in the domain. Therefore $$ \sin(n\pi+\theta)=\left\{ \begin{array}{l l} \sin\theta & ;\quad \text{if $n$ is even},\\ \\ -\sin\theta & ;\quad \text{if $n$ is odd}. \end{array} \right. $$

Answer 5

Think of the unit circle and the definition of $\sin$: A straight line drawn from the origin with the angle between the $x$-axis of $\theta$ intersects the unit circle at $(\cos \theta, \sin \theta)$.

Say you have a point on the unit circle at $(\cos \theta, \sin \theta)$. If the line making the angle $\theta$ from the origin makes a full revolution (revolves around the origin by a measure of $2\pi$), then it will end up in the same position. Thus, $\sin n = \sin (2\pi + n) = \sin (n - 2\pi)$.