Asked by Kris
1/Xin(x)
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Answered by
GPT-5 mini
AI
Do you mean 1/sin(x) (i.e., the cosecant function csc(x)) or do you mean something else (e.g., a named function Xin(x))? I’ll assume you meant 1/sin(x); tell me if that’s wrong.
Quick facts about 1/sin(x) = csc(x):
- Definition: csc x = 1/sin x.
- Domain: x ∈ R \ {nπ : n ∈ Z} (sin x = 0 excluded).
- Range: (-∞, -1] ∪ [1, ∞).
- Period: 2π. Parity: odd function (csc(-x) = -csc x).
- Vertical asymptotes at x = nπ.
- Derivative: d/dx[csc x] = -csc x cot x = -cos x / sin^2 x.
- Indefinite integral: ∫csc x dx = ln |tan(x/2)| + C = ln |csc x - cot x| + C.
- Series (Laurent) near x = 0: csc x = 1/x + x/6 + 7x^3/360 + O(x^5).
If you intended something else (e.g., derivative, integral, simplify, plot, or a different function named Xin), say what you want and I’ll continue.
Quick facts about 1/sin(x) = csc(x):
- Definition: csc x = 1/sin x.
- Domain: x ∈ R \ {nπ : n ∈ Z} (sin x = 0 excluded).
- Range: (-∞, -1] ∪ [1, ∞).
- Period: 2π. Parity: odd function (csc(-x) = -csc x).
- Vertical asymptotes at x = nπ.
- Derivative: d/dx[csc x] = -csc x cot x = -cos x / sin^2 x.
- Indefinite integral: ∫csc x dx = ln |tan(x/2)| + C = ln |csc x - cot x| + C.
- Series (Laurent) near x = 0: csc x = 1/x + x/6 + 7x^3/360 + O(x^5).
If you intended something else (e.g., derivative, integral, simplify, plot, or a different function named Xin), say what you want and I’ll continue.
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