C3 Trig

How do I solve sin(3X-53.1) = -1 without using cast? This was my solution but the answer is 107.7.

sin(3X-53.1)=-1
inverse sin both sides: 3x-53.1 = -90, 3x = -36.9, x=-12.3

Use general solutions.

$\sin(x)=\alpha \Rightarrow x=360n+\arcsin(\alpha), x=360n+180-\arcsin(\alpha), \forall n \in \mathbb{Z}$

How do I solve sin(3X-53.1) = -1 without using cast? This was my solution but the answer is 107.7.

sin(3X-53.1)=-1
inverse sin both sides: 3x-53.1 = -90, 3x = -36.9, x=-12.3

sin 270 also equals -1 so if you use that, using the same method, you get 107.7

Ah I see - is that something i'd just have to know?

I don't quite understand what you did. Why did you include the 360n and 180?

There is usually a range given in the question e.g. 0 < x < 360

but because you are solving 3x - 53 you need to adjust the range.

0 < 3x < 1080 multiplying inequality by 3,

-53 < x < 1027 so use that for CAST.

They're to do with symmetry of sine.

If you think about your problem, you essentially have a horizontal line intersecting some curve of sine. If you imagine you have $\frac{1}{2}=\sin(x)$ then if you were to sketch the line and the curve between 0 and 2pi then you would see that the line intersects the sine curve at $x=\arcsin(1/2)$ as well as some point later on. Due to symmetry of sine around $x=\frac{\pi}{2}$, you get that other point of intersection by going $\pi$ radians along the x-axis from 0, then turning around and going $-\arcsin(1/2)$ radians. So your two solutions here would be $x=\arcsin(1/2)$ and $x=\pi-\arcsin(1/2)$.

Since sine repeats the same pattern every $2\pi$, we can generalise by saying intersections happen every $2\pi$ which turns our equations into $x=2\pi n + \arcsin(1/2)$ and $x=2\pi n + \pi - \arcsin(1/2)$ where of course n can be any integer as it determines which period of sine you are referring to. You should choose it accordingly to your range.

Anyway, that's just a stupid long explanation for something very simple, and avoids the nonsense of CAST. Applies in the same way for degrees using conversion.

They're to do with symmetry of sine.

If you think about your problem, you essentially have a horizontal line intersecting some curve of sine. If you imagine you have $\frac{1}{2}=\sin(x)$ then if you were to sketch the line and the curve between 0 and 2pi then you would see that the line intersects the sine curve at $x=\arcsin(1/2)$ as well as some point later on. Due to symmetry of sine around $x=\frac{\pi}{2}$, you get that other point of intersection by going $\pi$ radians along the x-axis from 0, then turning around and going $-\arcsin(1/2)$ radians. So your two solutions here would be $x=\arcsin(1/2)$ and $x=\pi-\arcsin(1/2)$.

Since sine repeats the same pattern every $2\pi$, we can generalise by saying intersections happen every $2\pi$ which turns our equations into $x=2\pi n + \arcsin(1/2)$ and $x=2\pi n + \pi - \arcsin(1/2)$ where of course n can be any integer as it determines which period of sine you are referring to. You should choose it accordingly to your range.

Anyway, that's just a stupid long explanation for something very simple, and avoids the nonsense of CAST. Applies in the same way for degrees using conversion.

Your method is overly complicated for A level Maths where general solutions are NOT required - there is nothing wrong with CAST ...

Well he DID ask for a method without CAST so I provided one for him, it isn't too hard to understand. I find CAST as too much of a fuss as opposed to general solutions.

Why offer the general solution method then? It's not appropriate ...

It is a perfectly acceptable method within the A-Level exams as it is covered in FP1, I used it for C2, C3 and C4, so I'd say it is an appropriate substitute for CAST if OP understands it and prefers it.

It is not advised for A level Maths - it's not in the spec.

Just because you prefer it does not make it appropriate for everyone.

Just because it is not on the spec it does not make it inappropriate. I used a method in Mechanics 3 for a question in Further Pure 4, so obviously it was not on the mark scheme and I wasn't sure whether I would get the marks, and when my teacher reached out to AQA they said it is fine as long as it is clearly displayed and the correct solution is obtained. Same idea applies here.

Yes I prefer it, and I put it forward for someone who may prefer it as well, I'm not forcing them to use it.

It requires memorising the formulas for general solution - not a great idea.

Please don't over-complicate methods for people ... you could have suggested they talk to their teacher who will know better methods than your approach.

Memorising formulae is not difficult, they are not difficult to derive either.

This is not an over-complicated method, I simply explained everything that is going on with the method, OP can feel free to ignore it, and as I said, he/she is not forced to use it. It's completely up to OP whether they choose to ignore it, or look more into it, I merely provide it with my own explanation.

You are assuming a massive amount - most people find memorising formulas hard.

If you are going to offer help then please think hard about whether the approach you are suggesting is appropriate for that poster.

You are assuming a massive amount - most people find memorising formulas hard.

If you are going to offer help then please think hard about whether the approach you are suggesting is appropriate for that poster.

General solutions are fine, you have to learn so many formulae anyway... and its another tool you can use so I dont see why this is a problem