What distinguishes these things from each other: mathematics, algebra, trigonometry, and calculus?

Question

And are there any others that should be listed with these that I've left out, and, if so, what distinguishes it/them?

Answer 1

mathematics is operations (plus, minus, divide...)

algebra is balances, often with replacement of discrete numbers with lettered substitutions (unknowns).

trig is the study of shapes described as mathematical relations and is the basis of land surveying.

calc is the study of the effect of minute differences, called differentials.

other main classes of numerical analysis used in engineering include digital (computer solutions), set theory (modern math) and matrix (multi-dimentional systems).

and for areas not suited to rigid scientific analysis, empirical solutions.

Answer 2

mathematics - manipulation of numbers - adding subtracting, etc.

- algebra - equations with unknowns.

- trigonometry - mathamatics associated with circules and angles.

- calculus - the nobel prize of mathamatics.  deals with situations of movement, and equations that cannot be solved normally (like dividing by 0).

Answer 3

Mathematics is the over all field of study that includes algebra, trigonometry, calculus, and other mathematical subjects.

Trigonometry is literally the study of right triangles. More realistically though, it's the branch of mathematics that deals with the use of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.

Calculus is the branch of mathematics that deals with rates of change, namely derivatives and integrals.

"Algebra" has a few different meanings.  But in this context, I assume you mean the math dealing with variables and their manipulation, along with general equation solving.

As for "others" you "left out", there are tons more branches of mathematics: statistics, geometry, game theory, number theory, topology...the list goes on.