Symmetrical and periodic nature of trig functions
We say an object has a symmetry property if an aspect of it remains essentially the same after the object has been transformed in some systematic way.
From the unit circle definitions of the sine and cosine functions, we see that the function values repeat at intervals of $2\pi$2π. We say that these functions have a period of $2\pi$2π, meaning the function value at some number $x$x is always the same as the value at $x+2\pi$x+2π. That is, $\sin\left(x+2\pi\right)=\sin x$sin(x+2π)=sinx and $\cos\left(x+2\pi\right)=\cos x$cos(x+2π)=cosx.
We say the sine and cosine functions are symmetrical under a translation by $2\pi$2π.
There are other important symmetries possessed by these two functions. Again, by looking at the unit circle diagram, we observe that
$\sin\left(-x\right)=-\sin x$sin(−x)=−sinx and
$\cos\left(-x\right)=\cos x$cos(−x)=cosx
Any function that has the property $f\left(-x\right)=-f\left(x\right)$f(−x)=−f(x) is called an odd function. Thus, sine is an odd function. When it is represented by means of a graph, one can see that the picture will look the same if the graph is rotated about the origin by $180^\circ$180°. This property is characteristic of odd functions.
Any function that has the property $f\left(-x\right)=f\left(x\right)$f(−x)=f(x), is called an even function. Thus, the cosine function is an even function. The graph of any even function is the same as its reflection about the vertical axis.
We can check that the tangent function is an odd function. We make use of the unit circle definition of the tangent function:
$\tan\left(-x\right)=\frac{\sin\left(-x\right)}{\cos\left(-x\right)}=\frac{-\sin x}{\cos x}=-\frac{\sin x}{\cos x}=-\tan x$tan(−x)=sin(−x)cos(−x)=−sinxcosx=−sinxcosx=−tanx
The tangent function also has translational symmetry. It has a period of $\pi$π. We can verify from the unit circle diagram or from the graphs of sine and cosine that $\sin\left(x+\pi\right)=-\sin x$sin(x+π)=−sinx and $\cos\left(x+\pi\right)=-\cos x$cos(x+π)=−cosx. This means that $\tan\left(x+\pi\right)=\frac{\sin\left(x+\pi\right)}{\cos\left(x+\pi\right)}=\frac{-\sin x}{-\cos x}=\tan x$tan(x+π)=sin(x+π)cos(x+π)=−sinx−cosx=tanx for all values of $x$x and this is the required condition.
Worked examples
Question 1
Examine the graph of $y=\sin x$y=sinx.
How long is one cycle of the graph?
State the $x$x values for which $\sin x=0$sinx=0, from $x=0$x=0 to $x=2\pi$x=2π inclusive.
State the first $x$x value for which $\sin x=0.5$sinx=0.5
Using the symmetry of the graph, for what other value of $x$x shown on the graph does $\sin x=0.5$sinx=0.5?
Using the symmetry of the graph, for what values of $x$x does $\sin x=-0.5$sinx=−0.5?
Question 2
Examine the graph of $y=\tan x+2$y=tanx+2.
How long is one cycle of the graph?
State the $x$x values for which $\tan x+2=2$tanx+2=2, from $x=-2\pi$x=−2π to $x=2\pi$x=2π inclusive.
Write all answers on the same line separated by commas.
State the first positive $x$x value for which $\tan x+2=3$tanx+2=3
Using the period of the graph, for what other values of $x$x between $x=-2\pi$x=−2π and $x=2\pi$x=2π does $\tan x+2=3$tanx+2=3?
Write all answers on the same line separated by commas.
For what values of $x$x between $x=-2\pi$x=−2π and $x=2\pi$x=2π does $\tan x+2=1$tanx+2=1?
Question 3
Examine the graph of $y=\cos x$y=cosx.
State the exact value of $\cos\frac{\pi}{6}$cosπ6.
Use the graph to determine all other values of $x$x between $x=-\pi$x=−π and $x=\pi$x=π for which $\cos x=\pm\frac{\sqrt{3}}{2}$cosx=±√32.