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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 shouldnâ€™t get to excited about it when it happens.