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Section 1.4 : Solving Trig Equations

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

Hint : Find all the solutions to the equation without regard to the given interval. The first step in this process is to isolate the sine (with a coefficient of one) on one side of the equation.
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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.