Benjamin H. answered • 11/16/21
Harvard Grad/Experienced Tutor in STEM, English, and Writing
This looks scary but, surprisingly, it simplifies a lot! The trick here is that we have a “sinx” term in both terms we are adding. This means we can factor out sin(x) to get: sin(x) * (5* cos(x) + 4) = 0.
Now comes another quirk: If we let sin(x) = a, and (5 * cos (x) + 4) = b, then we essentially have the equation: a * b =0. As a rule of thumb, when two factors multiply together to get 0, this means that at least one of the two factors is 0.
So let’s consider the case where: a) sin (x) = 0. This one, fortunately, is straightforward: sin (x) = 0 anytime x is a multiple of 180 degrees. So two angles that satisfy this equation are x = 0, x = 180.
However, we now have to consider the other case: b) 5 * cos (x) + 4 = 0. We can rearrange this equation to get: 5 * cos (x) = -4. Then, we divide by 5 on both sides to get: cos (x) = -4/5. This one isn’t as intuitive, so we have to take the arccos(-4/5) [inverse cosine function] to find that x = 143.1. However, this isn’t the only solution, as for every value such that -1 < y < 1, there are two possible values of x.
So what’s the second one?
If we go to our unit circle representation for sin and cos, we want another angle value like 143.1 degrees where the angle produces a point with an x coordinate (cosine corresponds to x, sine corresponds to y) of -4/5. You can find this angle if you reflect your angle over the x-axis, which gives you the last answer, x=216.1。
