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Trigonometry Equation - Notes, Concept and All Important Formula

TRIGONOMETRIC EQUATION

1. TRIGONOMETRIC EQUATION :

An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation.


2. SOLUTION OF TRIGONOMETRIC EQUATION :

A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation.

(a) Principal solution :- The solution of the trigonometric equation lying in the interval \([0,2 \pi]\).

(b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of \(\theta\) for which trigonometric functions have the same value. All such possible values of \(\theta\) for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation.


3. GENERAL SOLUTIONS OF SOME TRIGONOMETRICE EQUATIONS (TO BE REMEMBERED) :

  (a) If \(\sin \theta=0\), then \(\theta= n \pi, n \in I\) (set of integers)

  (b) If \(\cos \theta=0\), then \(\theta=(2 n+1) \frac{\pi}{2}, n \in I\)

  (c) If \(\tan \theta=0\), then \(\theta= n \pi, n \in I\)

  (d) If \(\sin \theta=\sin \alpha\), then \(\theta= n \pi+(-1)^{n} \alpha, n \in I\)

  (e) If \(\cos \theta=\cos \alpha\), then \(\theta=2 n \pi \pm \alpha, n \in I\)

  (f) If \(\tan \theta=\tan \alpha\), then \(\theta= n \pi+\alpha, n \in I\)

  (g) If \(\sin \theta=1\), then \(\theta=2 n \pi+\frac{\pi}{2}=(4 n +1) \frac{\pi}{2}, n \in I\)

  (h) If \(\cos \theta=1\) then \(\theta=2 n \pi, n \in I\)

  (i) If \(\sin ^{2} \theta=\sin ^{2} \alpha\) or \(\cos ^{2} \theta=\cos ^{2} \alpha\) or \(\tan ^{2} \theta=\tan ^{2} \alpha\)  then  \(\theta= n \pi \pm \alpha, n \in I\)

  (j) \(\sin ( n \pi+\theta)=(-1)^{ n } \sin \theta, n \in I\)

       \(\cos ( n \pi+\theta)=(-1)^{ n } \cos \theta, n \in I\)


4. GENERAL SOLUTION OF EQUATION a \(\sin \theta+ b \cos \theta= c\) :

Consider, a \(\sin \theta+b \cos \theta=c \ldots \ldots \ldots \ldots \ldots\) (i)

\(\therefore \frac{a}{\sqrt{a^{2}+b^{2}}} \sin \theta+\frac{b}{\sqrt{a^{2}+b^{2}}} \cos \theta=\frac{c}{\sqrt{a^{2}+b^{2}}}\)

equation (i) has the solution only if \(|c| \leq \sqrt{a^{2}+b^{2}}\)

let \(\frac{ a }{\sqrt{ a ^{2}+ b ^{2}}}=\cos \phi, \frac{ b }{\sqrt{ a ^{2}+ b ^{2}}}=\sin \phi \quad \& \phi=\tan ^{-1} \frac{ b }{ a }\)

by introducing this auxiliary argument \(\phi\), equation (i) reduces to

\((\theta+\phi)=\frac{c}{\sqrt{a^{2}+b^{2}}}\)

Now this equation can be solved easily.


5. GENERAL SOLUTION OF EQUATION OF FORM :

\(a_{0} \sin ^{n} x+a_{1} \sin ^{n-1} x \cos x+a_{2} \sin ^{n-2} x \cos ^{2} x+\ldots \)\(\ldots \ldots .+a_{n} \cos ^{n} x=0\)

\(a_{0}, a_{1}, \ldots \ldots . a_{n}\) are real numbers

Such an equation is solved by dividing equation both sides by \(\cos ^{n} x\).


6. IMPORTANT TIPS:

  (a) For equations of the type \(\sin \theta= k\) or \(\cos \theta= k\), one must check that \(| k | \leq 1\)

  (b) Avoid squaring the equations, if possible, because it may lead to extraneous solutions.

  (c) Do not cancel the common variable factor from the two sides of the equations which are in a product because we may loose some solutions.

  (d) The answer should not contain such values of \(\theta\), which make any of the terms undefined or infinite.

  (e) Check that denominator is not zero at any stage while solving equations.

  (f) (i) If \(\tan \theta\) or \(\sec \theta\) is involved in the equations, \(\theta\) should not be odd multiple of \(\frac{\pi}{2}\).

       (ii) If \(\cot \theta\) or \(\operatorname{cosec} \theta\) is involved in the equation, \(\theta\) should not be integral multiple of \(\pi\) or 0 .

  (g) If two different trigonometric ratios such as \(\tan \theta\) and \(\sec \theta\) are involved then after solving we cannot apply the usual formulae for general solution because periodicity of the functions are not same.

  (h) If L.H.S. of the given trigonometric equation is always less than or equal to \(k\) and \(RHS\) is always greater than \(k\), then no solution exists. If both the sides are equal to \(k\) for same value of \(\theta\), then solution exists and if they are equal for different value of \(\theta\), then solution does not exist.

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