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Section 1.6 : Solving Trig Equations with Calculators, Part II

4. Find all the solutions to \(\displaystyle 3\cos \left( {\frac{{3y}}{7}} \right)\sin \left( {\frac{y}{2}} \right) + 14\cos \left( {\frac{{3y}}{7}} \right) = 0\). Use at least 4 decimal places in your work.

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Hint : Factor the equation and using basic algebraic properties get two equations that can be dealt with using known techniques.
Start Solution

Notice that each term has a cosine in it and so we can factor this out of each term to get,

\[\cos \left( {\frac{{3y}}{7}} \right)\left( {3\sin \left( {\frac{y}{2}} \right) + 14} \right) = 0\]

Now, we have a product of two factors that equals zero and so by basic algebraic properties we know that we must have,

\[\cos \left( {\frac{{3y}}{7}} \right) = 0\hspace{0.25in}{\rm{OR}}\hspace{0.25in}3\sin \left( {\frac{y}{2}} \right) + 14 = 0\]
Hint : Solve each of these two equations to attain all the solutions to the original equation.
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Each of these equations are similar to equations solved in the previous section. Therefore, we will be assuming that you can recall the solution process for each and we will not be putting in as many details. If you are unsure of the process you should go back to the previous section and work some of the problems there before proceeding with the solution to this problem.

We’ll start with,

\[\cos \left( {\frac{{3y}}{7}} \right) = 0\]

From a unit circle we can see that we must have,

\[\frac{{3y}}{7} = \frac{\pi }{2} + 2\pi n\hspace{0.25in}{\rm{OR}}\hspace{0.25in}\frac{{3y}}{7} = \frac{{3\pi }}{2} + 2\pi n\hspace{0.25in}\,\,n = 0, \pm 1, \pm 2, \ldots \]

Notice that we can further reduce this down to,

\[\frac{{3y}}{7} = \frac{\pi }{2} + \pi n\hspace{0.25in}\,\,n = 0, \pm 1, \pm 2, \ldots \]

Finally, the solutions from this equation are,

\[y = \frac{{7\pi }}{6} + \frac{{7\pi n}}{3}\hspace{0.25in}\,\,n = 0, \pm 1, \pm 2, \ldots \]

The second equation will take a little more (but not much more) work. First, isolating the sine gives,

\[\sin \left( {\frac{y}{2}} \right) = - \frac{{14}}{3} < - 1\]

At this point recall that we know \( - 1 \le \sin \theta \le 1\) and so this equation will have no solutions.

Therefore, the only solutions to this equation are,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = \frac{{7\pi }}{6} + \frac{{7\pi n}}{3}\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots }}\]

Do get too excited about the fact that we only got solutions from one of the two equations we got after factoring. This will happen on occasion and so we need to be ready for these cases when they happen.

If an interval had been given we would next proceed with plugging in values of \(n\) to determine which solutions fall in that interval. Since we were not given an interval this is as far as we can go.

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.