Cos(arcsinx)
| Input |
|---|
| cos(sin^(-1)(x)) |
| Result |
| sqrt(1 – x^2) |
| Alternate form |
| sqrt(1 – x) sqrt(x + 1) |
| Series expansion at x=-1 |
| sqrt(2) sqrt(x + 1) – (x + 1)^(3/2)/(2 sqrt(2)) – (x + 1)^(5/2)/(16 sqrt(2)) + O((x + 1)^(7/2)) (Puiseux series) |
| Series expansion at x=0 |
| 1 – x^2/2 – x^4/8 + O(x^5) (Taylor series) |
| Series expansion at x=1 |
| sqrt(2 – 2 x) + (sqrt(1 – x) (x – 1))/(2 sqrt(2)) – (sqrt(1 – x) (x – 1)^2)/(16 sqrt(2)) + (sqrt(1 – x) (x – 1)^3)/(64 sqrt(2)) – (5 sqrt(1 – x) (x – 1)^4)/(1024 sqrt(2)) + O((x – 1)^5) (generalized Puiseux series) |
| Series expansion at x=∞ |
| sqrt(-x^2) – sqrt(-x^2)/(2 x^2) + O((1/x)^4) (Puiseux series) |
| Derivative |
| d/dx(cos(sin^(-1)(x))) = -x/sqrt(1 – x^2) |
| Indefinite integral |
| integral sqrt(1 – x^2) dx = 1/2 (sqrt(1 – x^2) x + sin^(-1)(x)) + constant |
| Global maximum |
| max{cos(sin^(-1)(x))} = 1 at x = 0 |
| Definite integral |
| integral_(-1)^1 sqrt(1 – x^2) dx = pi/2~~1.5708 |