Calculus I - Solving Trig Equations

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Section 1.4 : Solving Trig Equations

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8. Without using a calculator find the solution(s) to \(16 = - 9\sin \left( {7x} \right) - 4\) that are in \(\displaystyle \left[ { - 2\pi ,\frac{{9\pi }}{4}} \right]\).

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Isolating the sine (with a coefficient of one) on one side of the equation gives,

\[\sin \left( {7x} \right) = - \frac{{20}}{9} < - 1\]

Okay, at this point we can stop all work. We know that \( - 1 \le \sin \theta \le 1\) for any argument and so in this case there is no solution. This will happen on occasion and we shouldn’t get to excited about it when it happens.