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There are three types of trigonometric functions, they are: 1- Plane Trigonometric Functions 2- Inverse Trigonometric Functions and 3- Hyperbolic Trigonometric Functions
Euler published the formula, which relates complex exponentials to trigonometric functions in 1748. See related link.
With ease, I suppose. The question depends on what you consider easy trigonometric functions.
Vectors.
Trigonometric identities involve certain functions of one or more angles. These identities are useful whenever expressions involving trigonometric functions need to be simplified.