Section 1.5 : Solving Trig Equations with Calculators, Part I
3. Find all the solutions to \(\displaystyle 3 = 6 - 11\sin \left( {\frac{t}{8}} \right)\). Use at least 4 decimal places in your work.
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Isolating the sine (with a coefficient of one) on one side of the equation gives,
\[\sin \left( {\frac{t}{8}} \right) = \frac{3}{{11}}\]First, using our calculator we can see that,
\[\frac{t}{8} = {\sin ^{ - 1}}\left( {\frac{3}{{11}}} \right) = 0.2762\]Now we’re dealing with sine in this problem and we know that the \(y\)-axis represents sine on a unit circle and so we’re looking for angles that will have a \(y\) coordinate of \(\frac{3}{{11}}\). This means that we’ll have angles in the first (this is the one our calculator gave us) and second quadrant. Here is a unit circle for this situation.
From the symmetry of the unit circle we can see that \(\pi - 0.2762 = 2.8654\) is the second angle.
From the discussion in the notes for this section we know that once we have these two angles we can get all possible angles by simply adding “\( + \,2\pi n\) for \(n = 0, \pm 1, \pm 2, \ldots \)” onto each of these.
This then means that we must have,
\[\frac{t}{8} = 0.2762 + 2\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}\frac{t}{8} = 2.8654 + 2\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots \]Finally, to get all the solutions to the equation all we need to do is multiply both sides by 8.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{t = 2.2096 + 16\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}t = 22.9232 + 16\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots }}\]Note that depending upon the amount of decimals you used here your answers may vary slightly from these due to round off error. Any differences should be slight and only appear around the 4th decimal place or so however.