Why does the basic trigonometric identity always work?

Question

Geometry

In the 8th grade, we went through the basics of trigonometry, now I'm in 9th, and I had this question: how to find the sine, cosine and tangent of an arbitrary non-rectangular triangle, I used the theorem

The square of each side of a triangle is equal to the sum of the squares of the other two sides without doubling the product of these sides by the cosine of the angle between them

From here I easily derived the cosine formula, okay, but how to find the sine? I remembered the Basic Trigonometric Identity (Further: GTT).

I googled whether it is correct to use it in this case. Yes, everything is ok, it turned out like this:
(Of course, you can open the brackets, but then the formula will be ugly and long)

But now I’m thinking: why does OTT work here if it is proved using the Pythagorean theorem?

Answer 1

> why does OTT work here if it is proved using the Pythagorean theorem?
From the order in which you apply the proved theorems, their truth does not change. OTT - proven for any angle value. Everything, regardless of whether you are counting something in a triangle, or the matrix of turning the plane by a given angle, or counting the area of ​​your piece of the pie.
OTT always applies! For any arguments. But, perhaps this is what confused you, after applying OTT and taking the root, in theory, you will get +-sqrt(...). Because x^2=9 => x=3 or -3. Two options for extracting the root, 2 options for the sinus.
But since you know that an angle in a triangle cannot exceed 180 degrees, you know that the sine of that angle is always non-negative. Therefore, -sqrt() can be discarded as an extra value and get your formula. In a formal proof, these considerations must be taken into account.