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begin{document}
tableofcontents
chapter{Graph of Trigonometric and Inverse Trigonometric Functions and Solution of Trigonometric Equations}
Trigonometric functions are usually defined either with the help of unit circle or right angled triangles. We also study their properties with a special emphasis on their graphs.
section{Introduction}
The domain of trigonometric functions are the set of angles, rather than real numbers. We can however, make the domains of
trigonometric functions, subsets of real numbers, by defining them on the unit circle, that is a circle whose radius is 1.
Let $theta$ be a central angle of the unit circle and $P(x,y)$ be the point as show in figure, then $r=OP=sqrt{x^2+y^2}$ and the six textbf{trigonometric functions} or textbf{circular functions} are defined as:
begin{center}
$sintheta=dfrac{y}{1}=y$
$costheta=dfrac{x}{1}=x$
$tantheta=dfrac{y}{x},$quad$xneq0$
$cosectheta=dfrac{1}{y},$ quad$yneq0$
$sectheta=dfrac{1}{x},$ quad$xneq0$
$cottheta=dfrac{x}{y},$ quad$yneq0.$
end{center}
begin{center}
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end{center}
Since any real number can represent the
length of exactly one arc on the unit circle. If $t$ is a positive number, we can find the arc of length $t$ by measuring a distance $t$ counterclockwise direction along an arc of the unit circle beginning at $C(1,0).$ So, we get arc $CP$ of length $t.$
If $t$ is negative number, we can find the arc of length $t,$ by measuring a distance $t$ clockwise direction an arc of the unit circle beginning at the point $C(1,0).$ In each case we get a unique point $P(x,y)$ that corresponds to a real number $t.$
We also know if $s$ is an arc which subtends an angle $theta$ at the center of circle with radius $r,$ we have $s=rtheta$ where $theta$ is in radians.
Let $s=t$ and $r=1$ then above equation reduce to $t=theta$ or $theta=t.$
Thus we obtain:
begin{center}
$sintheta=sin t,$quadquad
$costheta=cos t$
$tantheta=tan t$quadquad
$cosectheta=cosec t$
$sectheta=sec t$quadquad
$cottheta=cot t.$
end{center}
Thus we can think of each trigonometric expression as being either a trigonometric function of an angle measured in radians or as a trigonometric function of a real number $t.$
Thus the trigonometric functions can be thought of as functions that have domains and ranges that are subsets of real numbers.
section{Domain and Range of Trigonometric Functions}
subsection*{Domain and Range of Sine and Cosine Functions}
We know from above discussion that
$sintheta=y$quadquadquad$costheta=x$
Domain of sine and cosine the set of real numbers $mathbb{R}.$
Since any $P(x,y)$ is on the unit circle
$therefore$quadquad$-1leq yleq1$quadquad andquad$-1leq xleq1.$
orthinspace$-1leq sinthetaleq1$quad andquad$-1leq costhetaleq1.$
Thus the range of sine and cosine functions are $[-1,1].$
subsection*{Domain and Range of Tangent and Cotangent Functions}
We know that:
$tantheta=dfrac{y}{x},$ quadquad$xneq0.$
When $x=0,$ then terminal side can not coincide with $OY$ or $acute{OY};$ in other words
$thetaneq pmdfrac{pi}{2},pmdfrac{3pi}{2},pmdfrac{5pi}{2},...$ or $thetaneqdfrac{(2n+1)pi}{2};ninmathbb{Z}.$
$therefore$quad Domain=$mathbb{R}-biggl{t|t=(2n+1)dfrac{pi}{2};ninmathbb{Z}biggr}.$
and Range=$mathbb{R}(text{Set of real numbers}.)$
Since quad$cottheta=dfrac{x}{y}, quad$ $yneq0$
when $yneq0$ then terminal side OP does not coincide with $OX$ or $acute{OX},$ in other words
$thetaneq0 pmpi,pm2pi,pm3pi,...$ or
$thetaneq npi;quad ninmathbb{Z}.$
$therefore$quad Domain=$mathbb{R}-biggl{t|t=npi;ninmathbb{Z}biggr}.$
and Range=$mathbb{R}(text{Set of real numbers}).$
subsection*{Domain and Range of Secant and Cosecant Functions}
Since quad$cosectheta=dfrac{1}{y}, quad yneq0$
If $yneq0,$ then as we seen in case of $cottheta, quad$
$thetaneq npi;quad ninmathbb{Z}.$
$therefore$quad Domain=$mathbb{R}-{t|t=npiquad;ninmathbb{Z}}.$
Since $|y|=sqrt{y^2}leqsqrt{x^2+y^2}=1$
$Rightarrow$$dfrac{1}{|y|}leq1.$ Thus either $dfrac{1}{y}geq1$ or $dfrac{1}{y}leq-1$ that is $cosecthetageq1$ or $cosecthetaleq-1$
That is $cosectheta$ attains all values except those that lie between $-1$ and $1.$
Hence Range=$mathbb{R}-{t|tin(-1,1)}.$
Now we know that $sectheta=dfrac{1}{x}quad$ $xneq0,$ thus as seen in case of $tantheta$
$thetaneq(2n+1)dfrac{pi}{2}quad ninmathbb{Z}.$
$therefore$quad Domain=$mathbb{R}-bigg{t|t=(2n+1)dfrac{pi}{2};ninmathbb{Z}biggr}.$
Also $|x|=sqrt{x^2}leqsqrt{x^2+y^2}=1$
$Rightarrow$$|x|leq1$
$Rightarrow$$dfrac{1}{|x|}geq1$
Thus either $dfrac{1}{x}geq1$ or $dfrac{1}{x}leq-1$ that is
$secthetageq1$ or $secthetaleq-1.$
That is $sectheta$ attains all values except those which lie between $-1$ and $1.$
$therefore$quad Range=$mathbb{R}-{t|tin(-1,1)}.$
begin{ex}
Find the domain of the each of the following functions.
textbf{(i)}quad$sec3x$quadtextbf{(ii)} quad$tandfrac{1}{5}x$quadtextbf{(iii)}quad$cosecdfrac{1}{2}x.$
textbf{Solution:} textbf{(i)} We know that the domain of $sec t$ is:quad $-infty<t<infty,$ quad $tneq(2n+1)dfrac{pi}{2},thinspace ninmathbb{Z}.$quad
If $t=3x,$ then the domain
$-infty<3x<infty,$ quad $3xneq(2n+1)dfrac{pi}{2},thinspace ninmathbb{Z}$
$Rightarrow$$-infty<x<infty$ thinspace$xneq(2n+1)dfrac{pi}{6};quad ninmathbb{Z}.$
$therefore$ Domain of $sec3x=mathbb{R}-bigg{t|t=(2n+1)dfrac{pi}{6}bigg}.$
textbf{(ii)} Domain of $tan t$ is:
$-infty<t<infty,$ quad $tneq(2n+1)dfrac{pi}{2},quad ninmathbb{Z}.$ If $t=dfrac{1}{5}x$ then dom of $tandfrac{1}{5}x$ is
$-infty<dfrac{1}{5}x<infty,$quad$dfrac{1}{5}xneq(2n+1)dfrac{pi}{2}$
$Rightarrow$ $-infty<x<infty$quad $xneq(2n+1)dfrac{5pi}{2}.$
$therefore$ Domain of {small $tandfrac{1}{5}x=mathbb{R}-bigg{x|x=5(2n+1)dfrac{pi}{2}bigg}.$}
textbf{(iii)} Domain of $cosec t$ is $-infty<t<infty,$quad$tneq npiquad ninmathbb{Z}.$
Let $t=dfrac{1}{x}$ then domain of $cosecdfrac{1}{2}x$ is: $-infty<dfrac{1}{2}x<infty,$quad$dfrac{1}{2}xneq npiquad ninmathbb{Z}.$
$Rightarrow$$-infty<x<infty,$quad$xneq2npiquad ninmathbb{Z}.$
$therefore$quad Domain of $cosecdfrac{1}{2}x
=mathbb{R}-{x|x=2npi;ninmathbb{Z}}.$
end{ex}
begin{ex}
Find the range of the each function.
textbf{(i)}quad $cos3x$quadtextbf{(ii)}quad$3tan2x$quadtextbf{(iii)} $2cosecdfrac{1}{3}x$
textbf{Solution:} textbf{(i)} We know that for all
$tin text{domain of}cos t,$
$-1leqcos tleq1.$
Let $t=3x$ then $-1leqcos3xleq1.$
Hence range of $cos3x$ is the closed interval $[-1,1].$
textbf{(ii)} Since for all $t$ in domain of $tan t,$quad
$-infty<tan t<infty.$
Let $t=2x$ then
$-infty<tan2x<infty$
$Rightarrow$$-infty<3tan2x<infty.$
Thus the range of $3tan2x$ is $mathbb{R}.$
textbf{(iii)} Since for all $t$ in domain of $cosec t$
$cosectleq-1$ or $cosectgeq1.$
Let $t=dfrac{1}{3}x$ then
$cosecdfrac{1}{3}xleq-1$ or $cosecdfrac{1}{3}xgeq1$
Hence $2cosecdfrac{1}{3}xleq-2$ or $2cosecdfrac{1}{3}xgeq2.$
Hence the range of
$2cosecdfrac{1}{3}x=mathbb{R}-{p|pin(-2,2)}.$
end{ex}
section*{Even and Odd Trigonometric Functions}
textbf{(a)} A function $f(x)$ is said to be even
ifquad $f(-x)=f(x).$
textbf{(b)} A function $f(x)$ is said to be odd
ifquad $f(-x)=-f(x).$
textbf{Note:} textbf{(i)} Trigonometric functions are either even or odd.
The trigonometric functions $sintheta, cosectheta, tantheta, cottheta$ are odd while $costheta$ and $sectheta$ are even functions.
textbf{(ii)} The sum of an odd function and an even function is neither even nor odd.
section{Periodicity of Trigonometric Functions}
A function $f(x)$ is said to be periodic if there exists a smallest positive number $k$ such thatquad
$f(x+k)=f(x),$
the smallest such positive number is called period of the function $f(x).$
textbf{Note:} All six trigonometric functions are periodic functions, because they repeat their values after their periods. This behaviour of trigonometric functions is called periodicity.
end{document}
The name of the chapter is too long. It is mixing with pages numbering and also mixing with the section in the chapter. How fix this problem!
The sectioning commands (such as chapter
) accept an optional argument. If you give the optional argument, then that is what appears in the TOC and header: so with chapter[short]{long title}
, you'd get "short" in the TOC and header, but "long title" as the chapter title.
Correct answer by Teepeemm on September 27, 2021
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