Here, you will learn what is trigonometric equation and how to find general solution of trigonometric equation with examples.
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An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometrical equation.
Solution of Trigonometric Equation
A value of the unknown angle which satisfies the given equation 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 :-ย Since all the trigonometric functions are many one & periodic, hence there are infinite values values of 0 for which trigonometric functions have the same value. All such possible values of 0 for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solution of trigonometric equation.
(c) Particular Solution :- The solution of the trigonometric equation lying in the given interval.
General Solution of Trigonometric Equation
(a) If sin \(\theta\) = 0, then \(\theta\) = n\(\pi\), n \(\in\) I (set of integers)
(b) If cos \(\theta\) = 0, then \(\theta\) = (2n+1)\(\pi\over 2\), n \(\in\) I
(c) If tan \(\theta\) = 0, then \(\theta\) = n\(\pi\), n \(\in\) I
(d)ย If cot \(\theta\) = 0, then \(\theta\) = (2n+1)\(\pi\over 2\), n \(\in\) I
Note : Since sec \(\theta\) \(\ge\) 1 or sec \(\theta\) \(\le\) 1, therefore sec \(\theta\) = 0 does not have any solution.
Similarly, cosec \(\theta\) = 0 has no solution.
Example : Find the general solution of trigonometric equation :
(i) \(sin2\theta\) = 0
(ii) \(tan{3\theta\over 4}\) = 0
Solution :
(i) \(sin2\theta\) = 0
\(\implies\) \(2\theta\) = \(n\pi\), where n \(\in\) Z ย ย [sin \(\theta\) = 0, then \(\theta\) = n\(\pi\)]
\(\implies\) \(\theta\) = \(n\pi\over 2\), where n \(\in\) Z
(ii) \(tan{3\theta\over 4}\) = 0
\(\implies\) \(3\theta\over 4\) = \(n\pi\), where n \(\in\) Z ย ย [tan \(\theta\) = 0, then \(\theta\) = n\(\pi\)]
\(\implies\) \(\theta\) = \(4n\pi\over 3\), where n \(\in\) Z
(d) If sin \(\theta\) = sin \(\alpha\), then \(\theta\) = n\(\pi\) + \({(-1)}^n\alpha\), where \(\alpha\) \(\in\) [-\(\pi\over 2\), \(\pi\over 2\)], n \(\in\) I
(e) cos \(\theta\) = cos \(\alpha\), then \(\theta\) = 2n\(\pi\) \(\pm\) \(\alpha\), where \(\alpha\) \(\in\) [0,\(\pi\)], n \(\in\) I
(f) tan \(\theta\) = tan \(\alpha\), then \(\theta\) = n\(\pi\) + \(\alpha\), where \(\alpha\) \(\in\) (-\(\pi\over 2\), \(\pi\over 2\)), n \(\in\) I
Example : Find the general solution of trigonometric equation :
(i) \(sin\theta\) = \(\sqrt{3}\over 2\)
(ii) \(cos3\theta\) = \(-1\over 2\)
Solution :
(i) A value of \(\theta\) satisfying \(sin\theta\) = \(\sqrt{3}\over 2\) is \(\pi\over 3\)
\(sin\theta\) = \(\sqrt{3}\over 2\)
\(\implies\) \(sin\theta\) = \(sin\pi\over 3\) \(\implies\) \(\theta\) = n\(\pi\) + \({(-1)}^n{\pi\over 3}\),
(ii) \(cos3\theta\) = \(-1\over 2\)
\(\implies\) \(cos3\theta\) = \(cos{2\pi\over 3}\) \(\implies\) \(3\theta\) = 2n\(\pi\) \(\pm\) \(2\pi\over 3\)
\(\implies\) \(\theta\) = 2n\(\pi\over 3\) \(\pm\) \(2\pi\over 9\)
(g) If sin \(\theta\) = 1, then \(\theta\) = 2n\(\pi\) + \(\pi\over 2\) = (4n + 1)\(\pi\over 2\), n \(\in\) I
(h)ย If cos \(\theta\) = 1, then \(\theta\) = 2n\(\pi\), 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
Example : Find the general solution of trigonometric equation \(7cos^2\theta\) + \(3sin^2\theta\) = 4
Solution : We have,
\(7cos^2\theta\) + \(3sin^2\theta\) = 4
\(7(1-sin^2\theta)\) + \(3sin^2\theta\) = 4
\(4sin^2\theta\) = 3
\(sin^2\theta\) = \(3\over 4\) = \(({\sqrt{3}\over 2})^2\)
\(sin^2\theta\) = \(sin^2{\pi\over 3}\) \(\implies\) \(\theta\) = \(n\pi\pm{\pi\over 3}\).
