Trigonometric Equation Examples

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.

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