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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Solving Trigonometric Equations


The various trigonometric formulae and identities can be used to help solve trigonometric equations. Here is a summary of the most important trigonometric formulae you should know:


\displaystyle \sin^2x + \cos^2x = 1


\displaystyle 1 + \cot^2x = \csc^2x


\displaystyle \tan^2x + 1 = \sec^2x


\displaystyle \cos2x = \cos^2x - \sin^2x = 2\cos^2x - 1 = 1 - 2\sin^2x


\displaystyle \sin2x = 2\sinx \cosx


\displaystyle \tan \x = \frac{\sin x}{\cos x}


Example

Solve \displaystyle 2\cos^2x + 3\sin x = 3, giving your answer in radians for 0 < x < p.


\displaystyle 2\cos^2x + 3\sin^x - 3 = 0

We need to get everything in terms of \sin x or everything in terms of \cos x. Since we know that \cos^2x = 1 - \sin^2x:

\displaystyle 2(1 - \sin^2x) + 3\sinx - 3 = 0

\displaystyle 2 - 2\sin^2x + 3\sinx - 3 = 0

\displaystyle -2\sin^2x + 3\sinx - 1 = 0

\displaystyle 2\sin^2x - 3\sinx + 1 = 0

\displaystyle (2\sinx - 1)(\sinx - 1) = 0

\displaystyle \sin x = \frac{1}{2} or \displaystyle \sin x = 1


\displaystyle x = \frac{p}{6},\ \frac{5p}{6},\ \frac{p}{2}.


Remember, if \displaystyle \sin x = 1,\  x = \frac{p}{2},\ \frac{5p}{2},\ \frac{9p}{2},... and the same is true for \displaystyle \arcsin (\frac{1}{2}).

In the question, you are asked for values of x between 0 and p. You must write down all of the appropriate solutions.


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