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Section 1.6 : Solving Trig Equations with Calculators, Part II

8. Find all the solutions to \(4y\sec \left( {7y} \right) = - 21y\). 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.
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Notice that if we move all the terms to one side we can then factor a \(y\) out of the equation. Doing this gives,

\[\begin{align*}4y\sec \left( {7y} \right) + 21y & = 0\\ y\left( {4\sec \left( {7y} \right) + 21} \right) & = 0\end{align*}\]

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

\[y = 0\hspace{0.25in}{\rm{OR}}\hspace{0.25in}4\sec \left( {7y} \right) + 21 = 0\]

Be careful with this type of equation to not make the mistake of just canceling the \(y\) from both sides in the initial step. Had you done that you would have missed the \(y = 0\) solution.

When solving equations it is important to remember that you can’t cancel anything from both sides unless you know for a fact that what you are canceling will never be zero.

Hint : Solve each of these two equations to attain all the solutions to the original equation.
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There really isn’t anything that we need to do with the first equation and so we can move right on to the second equation. Note that this equation is 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.

First, isolating the secant and converting to cosines (to make the solving a little easier) gives,

\[\sec \left( {7y} \right) = - \frac{{21}}{4}\hspace{0.25in} \to \hspace{0.25in}\cos \left( {7y} \right) = - \frac{4}{{21}}\]

Using our calculator we get,

\[7y = {\cos ^{ - 1}}\left( { - \frac{4}{{21}}} \right) = 1.7624\]

From a quick look at a unit circle we know that the second angle in the range \(\left[ {0,2\pi } \right]\) will be \(2\pi - 1.7624 = 4.5208\).

Finally, the solutions to this equation are,

\[\begin{align*}7y & = 1.7624 + 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}7y & = 4.5208 + 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \\y & = 0.2518 + \frac{{2\pi n}}{7} & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}y &= 0.6458 + \frac{{2\pi n}}{7} & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \end{align*}\]

Putting all of this together gives the following set of solutions.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = 0, \hspace{0.25in} y = 0.2518 + \frac{{2\pi n}}{7}, \hspace{0.25in} {\rm{OR }} \hspace{0.25in} y = 0.6458 + \frac{{2\pi n}}{7}\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots }}\]

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.