Here you will learn some trigonometric equation examples for better understanding of trigonometric equation concepts.
Example 1 : Find general solution of (2sinx – cosx)(1 + cosx) = \(sin^2x\)
Solution : (2sinx – cosx)(1 + cosx) – (1 – \(cos^2x\)) = 0
\(\therefore\) (1 + cosx)(2sinx – cosx – 1 + cosx) = 0
\(\therefore\) (1 + cosx)(2sinx – 1) = 0
\(\implies\) cosx = -1 or sinx = \(1\over 2\)
\(\implies\) cosx = -1 = cos\(\pi\) \(\implies\) x = 2n\(\pi\) + \(\pi\) = (2n+1)\(\pi\), n \(\in\) I
or sinx = \(1\over 2\) = sin\(\pi\over 6\) \(\implies\) x = k\(\pi + (-1)^k{\pi\over 6}\), k \(\in\) I
Example 2 : Solve : 6 – 10cosx = 3\(sin^2x\)
Solution : we have, 6 – 10cosx = 3\(sin^2x\)
\(\therefore\) 6 – 10cosx = 3 – 3\(cos^2x\)
\(\implies\) 3\(cos^2x\) – 10cosx + 3 = 0
\(\implies\) (3cosx-1)(cosx-3) = 0 \(\implies\) cosx = \(1\over 3\) or cosx = 3
Since cosx =3 is not possible as -1 \(\le\) cosx \(\le\) 1
\(\therefore\) cosx = \(1\over 3\) = cos(\(cos^{-1}{1\over 3}\)) \(\implies\) x = 2n\(\pi\) \(\pm\) \(cos^{-1}{1\over 3}\)
Example 3 : Solve : cos3x + sin2x – sin4x = 0
Solution : we have, cos3x + (sin2x – sin4x) = 0
= cos3x – 2sinx.cos3x = 0
\(\implies\) (cos3x)(1 – 2sinx) = 0
\(\implies\) cos3x = 0 or sinx = \(1\over 2\)
\(\implies\) cos3x = 0 = cos\(\pi\over 2\) or sinx = \(1\over 2\) = sin\(\pi\over 6\)
\(\implies\) 3x = 2n\(\pi\) \(\pm\) \(\pi\over 2\) or x = m\(\pi\) + \({(-1)}^m\)\(\pi\over 6\)
\(\implies\) x = \(2n\pi\over 3\) \(\pm\) \(\pi\over 6\) or x = m\(\pi\) + \({(-1)}^m\)\(\pi\over 6\); (n, m \(\in\) I)
Practice these given trigonometric equation examples to test your knowledge on concepts of trigonometric equation.
