sin(x), sin(2x), sin(1/2)....what are the periods?
Pls explain to me the concepts involved so I can remember more clearly.
Thanks.
A period is how long it takes for the wave to complete one full cycle.
Sin(x) has a period of 2$\displaystyle \pi$ in radians, or $\displaystyle 360^{\circ}$ in degrees. This means that if you draw the graph of sin(x), the graph would take 2$\displaystyle \pi$ or $\displaystyle 360^{\circ}$ to complete one full cycle. Which is a period.
Sin(2x) has a period of half of sin(x)
Sin(0.5x) has a period of double of sin(x)
The 2 in front of the x doubles the speed of one wave, which means that it halves the time, and since a period is the length of time for one full cycle, the period is halved as well. Vise versa for the 0.5.
Just think of it as this way:
$\displaystyle sin(x) = \text{Normal period}$
$\displaystyle sin(2x) = \frac{\text{Normal period}}{2}$
$\displaystyle sin(0.5x) = \frac{\text{Normal period}}{0.5}$
$\displaystyle \displaystyle\ Sin(0)=0, \; \; Sin\left(\frac{{\pi}}{2}\right)=1, \; \; Sin({\pi})=0,\; \; Sin\left(\frac{3{\pi}}{2}\right)=-1,\; \; Sin(2{\pi})=0$
$\displaystyle Sinx$ goes through a full period in $\displaystyle 2{\pi}$ radians.
Finding 3 consecutive zeros will discover the period, as the centreline of the graph is the x-axis.
Simplest is to start at zero.
$\displaystyle \displaystyle\ Sin(2x)=0$ for $\displaystyle \displaystyle\ x=0$ and $\displaystyle \displaystyle\ 2x={\pi}\Rightarrow\ x=\frac{{\pi}}{2}$ and $\displaystyle 2x=2{\pi}\Rightarrow\ x={\pi}$
therefore $\displaystyle Sin(2x)$ has period $\displaystyle {\pi}$ radians.
$\displaystyle \displaystyle\ Sin\left(\frac{x}{2}\right)=0\Rightarrow\ x=0, \;\; x=2{\pi}, \;\; x=4{\pi}$
so the period of $\displaystyle \displaystyle\ Sin\left(\frac{x}{2}\right)$ is $\displaystyle 4{\pi}$ radians.
Alternatively......
$\displaystyle SinA=Sin(A+2{\pi})$
$\displaystyle Sinx=Sin(x+2{\pi})$ has a period of $\displaystyle 2{\pi}$ radians
$\displaystyle Sin(2x)=Sin(2x+2{\pi})=Sin[2(x+{\pi})]$ which therefore has a period of $\displaystyle {\pi}$ radians
$\displaystyle \displaystyle\ Sin\left(\frac{x}{2}\right)=Sin\left(\frac{x}{2}+2 {\pi}\right)=Sin\left[\frac{1}{2}(x+4{\pi})\right]$ has a period of $\displaystyle 4{\pi}$ radians
just to add that if you have shifted signal ($\displaystyle \sin {x \pm A}$), left or right it will have same period but just moved to one or another side or if you multiply signal for example that "sin" ($\displaystyle A\sin {x}$) it will again have same period but just amplitude bigger or smaller depending by which number is multiplying (if it is more or less than 1)... same thing when looking at those signals of yours if it's variable (in this case "x") is multiplied with number bigger than 1 that means that it will change that many times faster than usual, and if is multiplied with number less than 1 it will go that many times slower than normal