By Courant, Hilbert
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However, unlike those functions, tan θ is unbounded and is discontinuous at the points θ = (2n + 1)π/2, for n = 0, ±1, ±2, . , where cos θ vanishes. Similarly, cotθ is discontinuous at the points where sin θ vanishes. 9. 1 we defined inverse functions. 9 that there are an infinite number of angles for a given value of sine, cosine or tangent. To obtain a single-valued function, we would therefore have to formally restrict the angular range of θ. 10c, respectively. 3, it would be natural to refer to these as sin−1 , cos−1 and tan−1 , but to avoid ambiguity with 1/sin, etc.
The ambiguity in the value of the polar angle corresponding to a given point can be removed by restricting the range of θ to 0 < θ < 2π. However, this is not always convenient. 5. 4 Polar angles θ, for constant r; diagrams (a) and (b) represent the same point P, and diagrams (c) and (d) represent the same point P . 21) together with the fact that the particle traverses a length of arc l = υt in time t. The angle increases indefinitely as t increases and θ = 2nπ + φ, with 0 < φ < 2π, corresponds to the particle arriving at the point (r, φ) after n complete revolutions since t = 0.
This definition is implicit, and for any x the function can be evaluated by first evaluating the polynomials P (i) (x), and then finding the roots of the resulting polynomial in y. For n = 1, one easily sees that the above definition reduces to a rational function, or a polynomial in the case of P (0) = 1. e. addition, subtraction, multiplication and division). In contrast, functions that are not of the above form cannot be defined by a finite sequence of basic algebraic operations. Such functions are called transcendental functions and are somewhat analogous to irrational numbers, which cannot be evaluated from integers by a finite sequence of the operations of arithmetic.