Question: How would you solve this sinusoidal equation:
Solve $5\cos(6x)+6=9$. Assume $n$ is an integer and the answers are in degrees.
$-8.86+n\cdot 60$
$-3.54+n\cdot 60$
$3.54+n\cdot 60$
$8.86+n\cdot 60$
$15.13+ n\cdot 360$
$126.87+n\cdot 360$
I'm sort of new to this. But I have tried to isolate the trigonometric parts, and I get$$\cos(6x)=\frac 35\tag{1}$$
But after this, I'm not sure what to do. Do I take the $\arccos$ of both sides? If so, what will $\arccos\frac 35$ evaluate to? I don't think it's going to be a "perfect" number such as $\dfrac \pi 3$.
Best Answer
I don't believe there is a "nice" way to do it. Anyway the multiple choices all being terminating decimals should give you that hint.
Starting from $\cos(6x)=\dfrac 35$, WolframAlpha gives $8.86^{\circ}$ as a solution (like you said just take $\arccos$ of both sides and divide by $6$).
Now you know if $\cos (6 \cdot 8.86) = \dfrac 35$, then $\cos (6 \cdot 8.86 + 360n) = \dfrac 35$, so $\cos [6(8.86 + 60n)] = \dfrac 35$
Therefore $x = 8.86 + 60n$ is the solution.
If you really want to do it calculator free you can, but I wouldn't reccoment it.
EDIT:
Since cosine is an even function (i.e. $\cos (-\theta) = \cos \theta )$, another family of solutions is $-8.86 - 60n$, or just $-8.86 + 60n$. So the question has two answers.