Trigonometrical equation | General solution |
\[\sin \theta =0\] | \[\theta =n\pi \] |
\[\cos \theta =0\] | \[\theta =n\pi +\pi /2\] |
\[\tan \theta =0\] | \[\theta =n\pi \] |
\[\sin \theta =1\] | \[\theta =2n\pi +\pi /2\] |
\[\cos \theta =1\] | \[\theta =2n\pi \] |
\[\sin \theta =\sin \alpha \] | \[\theta =n\pi +{{(-1)}^{n}}\alpha \] |
\[\cos \theta =\cos \alpha \] | \[\theta =2n\pi \pm \alpha \] |
\[\tan \theta =\tan \alpha \] | \[\theta =n\pi \pm \alpha \] |
\[{{\sin }^{2}}\theta ={{\sin }^{2}}\alpha \] | \[\theta =n\pi \pm \alpha \] |
\[{{\tan }^{2}}\theta ={{\tan }^{2}}\alpha \] | \[\theta =n\pi \pm \alpha \] |
\[{{\cos }^{2}}\theta ={{\cos }^{2}}\alpha \] | \[\theta =n\pi \pm \alpha \] |
\[\left. \begin{align} & \sin \theta =\sin \alpha \\ & \cos \theta =\cos \alpha \text{ } \\ \end{align} \right|\text{ * }\] | more...
An equation involving one or more trigonometrical ratio of an unknown angle is called a trigonometrical equation
i.e., \[\sin x+\cos 2x=1\],\[(1-\tan \theta )(1+\sin 2\theta )=1+\tan \theta \], \[|\sec \left( \theta +\frac{\pi }{4} \right)|\text{ }=2\] etc.
A trigonometric equation is different from a trigonometrical identities. An identity is satisfied for every value of the unknown angle e.g.,\[{{\cos }^{2}}x=1-{{\sin }^{2}}x\]is true \[\forall x\in R\], while a trigonometric equation is satisfied for some particular values of the unknown angle.
(1) Roots of trigonometrical equation : The value of unknown angle (a variable quantity) which satisfies the given equation is called the root of an equation, e.g., \[\cos \theta =\frac{1}{2}\], the root is \[\theta ={{60}^{o}}\] or \[\theta ={{300}^{o}}\] because the equation is satisfied if we put \[\theta ={{60}^{o}}\]or \[\theta ={{300}^{o}}\].
(2) Solution of trigonometrical equations : A value of the unknown angle which satisfies the trigonometrical equation is called its solution.
Since all trigonometrical ratios are periodic in nature, generally a trigonometrical equation has more than one solution or an infinite number of solutions. There are basically three types of solutions:
(i) Particular solution : A specific value of unknown angle satisfying the equation.
(ii) Principal solution : Smallest numerical value of the unknown angle satisfying the equation (Numerically smallest particular solution).
(iii) General solution : Complete set of values of the unknown angle satisfying the equation. It contains all particular solutions as well as principal solutions.
The function, \[f(x)\] is differentiable at point \[P,\] iff there exists a unique tangent at point \[P\]. In other words, \[f(x)\] is differentiable at a point \[P\] iff the curve does not have \[P\] as a corner point. i.e., "the function is not differentiable at those points on which function has jumps (or holes) and sharp edges.”
Let us consider the function \[f(x)=|x-1|\], which can be graphically shown,
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(1) Discontinuous function : A function \['f'\] which is not continuous at a point \[x=a\] in its domain is said to be discontinuous there at. The point \['a'\] is called a point of discontinuity of the function.
The discontinuity may arise due to any of the following situations.
(i) \[\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\] or \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\] or both may not exist
(ii) \[\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\]as well as \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\] may exist, but are unequal.
(iii)\[\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\] as well as \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\] both may exist, but either of the two or both may not be equal to \[f(a)\].
Function \[f(x)\] is said to be
(1) Left continuous at \[x=a\] if \[\underset{x\to {{a}^{-}}}{\mathop{\text{lim}}}\,f(x)=f(a)\]
(2) Right continuous at \[x=a\] if \[\underset{x\to {{a}^{+}}}{\mathop{\text{lim}}}\,f(x)=f(a)\].
Thus a function \[f(x)\] is continuous at a point \[x=a\] if it is left continuous as well as right continuous at \[x=a.\]
Properties of continuous functions : Let \[f(x)\] and \[g(x)\] be two continuous functions at \[x=a.\]Then
(i) A function \[f(x)\] is said to be everywhere continuous if it is continuous on the entire real line R i.e. \[(-\infty ,\infty )\]. e.g., polynomial function, \[{{e}^{x}},\]\[\sin x,\,\cos x,\,\]constant, \[{{x}^{n}},\] \[|x-a|\] etc.
(ii) Integral function of a continuous function is a continuous function.
(iii) If \[g(x)\] is continuous at \[x=a\] and \[f(x)\] is continuous at \[x=g(a)\] then \[(fog)\,(x)\] is continuous at \[x=a\].
(iv) If \[f(x)\] is continuous in a closed interval \[[a,\,\,b]\] then it is bounded on this interval.
(v) If \[f(x)\] is a continuous function defined on \[[a,\,\,b]\] such that \[f(a)\] and \[f(b)\] are of opposite signs, then there is atleast one value of \[x\] for which \[f(x)\] vanishes. i.e. if \[f(a)>0,\,\,f(b)<0\Rightarrow \,\exists \,\,c\,\,\in \,\,(a,\,\,b)\] such that \[f(c)\,=0\].
A function \[f(x)\] is said to be continuous at a point \[x=a\] of its domain if and only if it satisfies the following three conditions :
(1) \[f(a)\] exists. (\['a'\] lies in the domain of \[f\])
(2) \[\underset{x\to a}{\mathop{\lim }}\,\,f(x)\] exist i.e.\[\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\] or R.H.L. = L.H.L.
(3) \[\underset{x\to a}{\mathop{\lim }}\,f(x)=f(a)\] (limit equals the value of function).
Cauchy’s definition of continuity : A function \[f\] is said to be continuous at a point \[a\] of its domain \[D\] if for every \[\varepsilon >0\] there exists \[\delta >0\] (dependent on \[\varepsilon )\] such that \[|x-a|<\delta \] \[\Rightarrow |\,f(x)-f(a)|<\varepsilon .\]
Comparing this definition with the definition of limit we find that \[f(x)\] is continuous at \[x=a\] if \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] exists and is equal to \[f(a)\] i.e., if \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=f(a)=\underset{x\to a+}{\mathop{\lim }}\,f(x)\].
The word ‘continuous’ means without any break or gap. If the graph of a function has no break or gap or jump, then it is said to be continuous.
A function which is not continuous is called a discontinuous function. While studying graphs of functions, we see that graphs of functions \[\sin x\], \[x,\] \[\cos x\], \[{{e}^{x}}\] etc. are continuous but greatest integer function \[[x]\] has break at every integral point, so it is not continuous. Similarly \[\tan x,\,\,\cot x,\,\,\sec x\], \[\frac{1}{x}\] etc. are also discontinuous function.
Continuous function
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We shall divide the problems of evaluation of limits in five categories.
(1) Algebraic limits : Let \[f(x)\] be an algebraic function and \['a'\] be a real number. Then \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] is known as an algebraic limit.
(i) Direct substitution method : If by direct substitution of the point in the given expression we get a finite number, then the number obtained is the limit of the given expression.
(ii) Factorisation method : In this method, numerator and denominator are factorised. The common factors are cancelled and the rest outputs the results.
(iii) Rationalisation method : Rationalisation is followed when we have fractional powers (like \[\frac{1}{2},\frac{1}{3}\] etc.) on expressions in numerator or denominator or in both. After rationalisation the terms are factorised which on cancellation gives the result.
(iv) Based on the form when \[x\to \infty \] : In this case expression should be expressed as a function \[1/x\] and then after removing indeterminate form, (if it is there) replace \[\frac{1}{x}\] by 0.
(2) Trigonometric limits : To evaluate trigonometric limit the following results are very important.
(i) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sin x}{x}=1=\underset{x\to 0}{\mathop{\lim }}\,\,\frac{x}{\sin x}\]
(ii) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\tan x}{x}=1=\underset{x\to 0}{\mathop{\lim }}\,\,\frac{x}{\tan x}\]
(iii) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{\sin }^{-1}}x}{x}=1=\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{\sin }^{-1}}x}\]
(iv) \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x}{x}=1=\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{\tan }^{-1}}x}\]
(v) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sin {{x}^{0}}}{x}=\frac{\pi }{180}\]
(vi) \[\underset{x\to 0}{\mathop{\lim }}\,\,\cos x=1\]
(vii) \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{\sin (x-a)}{x-a}=1\]
(viii) \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{\tan (x-a)}{x-a}=1\]
(ix) \[\underset{x\to a}{\mathop{\lim }}\,{{\sin }^{-1}}x={{\sin }^{-1}}a,\,\,|a|\,\,\le 1\]
(x) \[\underset{x\to a}{\mathop{\lim }}\,\,{{\cos }^{-1}}\,x={{\cos }^{-1}}\,a;\,\,|a|\,\,\le 1\]
(xi) \[\underset{x\to a}{\mathop{\lim }}\,\,{{\tan }^{-1}}\,x={{\tan }^{-1}}a;\,\,-\infty <a<\infty \]
(xii) \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin x}{x}=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\cos x}{x}=0\]
(xiii) \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin \left( 1/x \right)}{\left( 1/x \right)}=1\]
(3) Logarithmic limits : To evaluate the logarithmic limits we use following formulae
(i) \[\log (1+x)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-............\text{to}\,\infty \] where \[-1<x\le 1\] and expansion is true only if base is e.
(ii) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\log (1+x)}{x}=1\]
(iii) \[\underset{x\to e}{\mathop{\lim }}\,\,{{\log }_{e}}x=1\]
(iv) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\log (1-x)}{x}=-1\]
(v) \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\log }_{a}}(1+x)}{x}={{\log }_{a}}e,\,a>0,\ne 1\]
(4) Exponential limits
(i) Based on series expansion
We use \[{{e}^{x}}=1+x+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+.............\infty \]
To evaluate the exponential limits we use the following results
(a) \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}=1\]
(b) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{a}^{x}}-1}{x}={{\log }_{e}}a\]
(c) \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{e}^{\lambda x}}-1}{x}=\,\lambda \,\,(\lambda \ne 0)\]
(ii) Based on the form \[{{1}^{\infty }}\] : To evaluate the exponential form \[{{1}^{\infty }}\] we use the following results.
(a) If \[\underset{x\to a}{\mathop{\lim }}\,\,f(x)=\underset{x\to a}{\mathop{\lim }}\,\,g(x)=0\], then
\[\underset{x\to a}{\mathop{\lim }}\,\,{{\{1+f(x)\}}^{1/g(x)}}\,\,=\,{{e}^{\underset{x\to a}{\mathop{\lim }}\,\,\frac{f(x)}{g(x)}}}\] or when \[\underset{x\to a}{\mathop{\lim }}\,\,f(x)=1\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)=\infty \].
Then \[\underset{x\to a}{\mathop{\lim }}\,{{\{f(x)\}}^{g(x)}}=\underset{x\to a}{\mathop{\lim }}\,\,{{[1+f(x)-1]}^{g(x)}}\]\[={{e}^{\underset{x\to a}{\mathop{\lim }}\,(f(x)-1)g(x)}}\]
(b) \[\underset{x\to 0}{\mathop{\lim }}\,{{(1+x)}^{1/x}}=e\]
(c) \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( more...
The following theorems are very useful for evaluation of limits if \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=l\] and \[\underset{x\to 0}{\mathop{\lim }}\,g(x)=m\] (\[l\] and \[m\] are real numbers) then
(1) \[\underset{x\to a}{\mathop{\lim }}\,(f(x)+g(x))=l+m\,\] (Sum rule)
(2) \[\underset{x\to a}{\mathop{\lim }}\,(f(x)-g(x))=l-m\] (Difference rule)
(3) \[\underset{x\to a}{\mathop{\lim }}\,(f(x).g(x))=l.m\] (Product rule)
(4) \[\underset{x\to a}{\mathop{\lim }}\,k\,\,f(x)=k.l\] (Constant multiple rule)
(5) \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{f(x)}{g(x)}=\frac{l}{m},m\ne 0\] (Quotient rule)
(6) If \[\underset{x\to a}{\mathop{\lim }}\,f(x)=+\infty \] or \[-\infty \], then \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{1}{f(x)}=0\]
(7) \[\underset{x\to a}{\mathop{\lim }}\,\log \{f(x)\}=\log \,\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}\]
(8) If \[f(x)\le g(x)\] for all \[x,\] then \[\underset{x\to a}{\mathop{\lim }}\,f(x)\le \underset{x\to a}{\mathop{\lim }}\,g(x)\]
(9) \[\underset{x\to a}{\mathop{\lim }}\,{{[f(x)]}^{g(x)}}={{\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}}^{\underset{x\to a}{\mathop{\lim }}\,g(x)}}\]
(10) If \[p\] and \[q\] are integers, then \[\underset{x\to a}{\mathop{\lim }}\,{{(f(x))}^{p/q}}={{l}^{p/q}},\] provided \[{{(l)}^{p/q}}\] is a real number.
(11) If \[\underset{x\to a}{\mathop{\lim }}\,f(g(x))=f(\underset{x\to a}{\mathop{\lim }}\,g(x))=f(m)\] provided \['f'\] is continuous at \[g(x)=m.\,\,e.g.\]\[\underset{x\to a}{\mathop{\lim }}\,\ln [f(x)]=\ln (l),\]only if \[l>0.\]
Let \[y=f(x)\] be a function of \[x\]. If at \[x=a,f(x)\] takes indeterminate form, then we consider the values of the function which are very near to \['a'\]. If these values tend to a definite unique number as \[x\] tends to \['a'\], then the unique number so obtained is called the limit of \[f(x)\] at \[x=a\] and we write it as \[\underset{x\to a}{\mathop{\lim }}\,f(x)\].
(1) Left hand and right hand limit : Consider the values of the functions at the points which are very near to \[a\] on the left of \[a\]. If these values tend to a definite unique number as \[x\] tends to \[a,\] then the unique number so obtained is called left-hand limit of \[f(x)\] at \[x=a\] and symbolically we write it as \[f(a-0)=\]\[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,\,f(x)=\]\[\,\underset{h\to 0}{\mathop{\lim }}\,\,f(a-h)\].
Similarly we can define right-hand limit of \[f(x)\] at \[x=a\] which is expressed as \[f(a+0)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\]\[=\underset{h\to 0}{\mathop{\lim }}\,f(a+h)\].
(2) Method for finding L.H.L. and R.H.L.
(i) For finding right hand limit (R.H.L.) of the function, we write \[x+h\] in place of \[x,\] while for left hand limit (L.H.L.) we write \[x-h\] in place of \[x\].
(ii) Then we replace \[x\] by \['a'\] in the function so obtained.
(iii) Lastly we find limit \[h\to 0\].
(3) Existence of limit : \[\underset{x\to a}{\mathop{\lim }}\,f(x)\,\,\]exists when,
(i) \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\] exist i.e. L.H.L. and R.H.L. both exists.
(ii) \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\] i.e. L.H.L. = R.H.L.
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