Does Sin, Cos, and Tan Only Apply to Right Triangles Because That’s How We Defined It?

Yes. These are all trigonometric functions. In their definitions, the concept of a right triangle is used. However, the terms themselves find use in all sorts of triangles, even quadrilaterals, and polygons. In short, everything that has triangles in them.

Sine, Cosine, and Tangent: Are Defined for Angles, Not Triangles

Sine, cosine, and tangent trigonometric functions are defined for angles, not for triangles. Of course, since all triangles have 3 angles, these functions are defined for all three angles of any triangle, whether it is a right triangle or not.

Right Triangles and Unique Function Values

Because in a right triangle we have unique definitions of the angles leading to unique values of the functions. You know all angles if you know one besides the right one. In other triangles the values were not a function of the described angle only but at least of one other angle as well.

Trigonometric Functions Defined Using Right Triangles

The definitions for trig functions that we are familiar with use right triangles: SOH-CAH-TOA. Sine of an angle is opposite side divided by hypotenuse, etc. Trigonometric functions work for all triangles but the formula might be a little different from what we are used to.

Law of Cosines and Pythagorean Theorem

Remember that the Law of Cosines tells us that (a^2 b^2 - 2ab cos C c^2), with sides (a), (b), and (c) and angles (A), (B), and (C). In the special case when we have a right triangle, (C 90) degrees, the Law of Cosines simplifies to:

(a^2 b^2 - 2ab cdot 0 c^2 ) [since (cos 90^circ 0)] [a^2 b^2 c^2]

The last line above is more recognizable by its usual name: the Pythagorean Theorem. So the Pythagorean Theorem used to relate the 3 sides of a right triangle is a result of the Law of Cosines.

Additional Applications and Beyond Right Triangles

Ved Gadge has given the answer correctly. No need to repeat. Initially, they are taught with the help of a right triangle as a first step. Further development is a great achievement of human-kind.

Nope, all the trigonometric functions are applicable for all angles from 0 to 360 (excluding some trigonometric functions for particular angles, e.g., (tan 90^circ) is undefined). If you learn more about trigonometry, you will understand that it is not just related to a right triangle, but it is related to a circle.