Inverse Trigonometric Functions (C)

The equation $\sin(x) = 1/2$ has many solutions, as you can tell from the fact that the line y = 1/2 intersects the graph of $y = \sin(x)$ at many places. However, if we restrict our attention to the portion of the graph of $y = \sin(x)$ for $x\in[-\pi/2,\pi/2]$ we see that there is only one such point of intersection. Indeed, for any number $u\in[-1,1]$ there is exactly one number $v\in[-\pi/2,\pi/2]$ so that $\sin(v) = u$. We denote this number v by $\arcsin(u)$. More generally, we define the function $\arcsin:[-1..1]\rightarrow[-\pi/2..\pi/2]$ by the rule

$$\arcsin(u) = v\;\;{\rm if}\;\;\sin(v) = u\;\;{\rm and}\;\;v\in[-\pi/2,\pi/2]$$

It should be noted that $\sin(\arcsin(u)) = u$ for every $u\in[-1,1]$ but $\arcsin(\sin(x)) = x$ only if $x\in[-\pi/2,\pi/2]$. The quantity arcsin(u) has the following geometric interpretation. Consider the circle x² + y² = 1 and a point on that circle with coordinates $(\sqrt{1-u^2},u)$. If $u \geq 0$ then $\arcsin(u)$ is the length of the counter clockwise arc on that circle from (1,0) to $(\sqrt{1-u^2},u)$. If $u\leq 0$ then $-\arcsin(u)$ is the length of the clockwise arc of the circle from (1,0) to $(\sqrt{1-u^2},u)$.

Similarly, we define the arccosine function, $\arccos$, by considering the graph of $y = \cos(x)$ for $x\in[0,\pi]$ and considering the solutions of $\cos(v) = u$ for $u\in[-1,1]$ and v restricted to $[0,\pi]$. Formally, the function $\arccos$ is defined by $\arcsin:[-1..1]\rightarrow[-\pi/2..\pi/2]$with the rule

$$\arccos(u) = v\;\;{\rm if}\;\;\cos(v) = u\;\;{\rm and}\;\;v\in[0,\pi]$$

The arccosine function has the following geometric interpretation. Consider the point $(u,\sqrt{1-u^2})$ on the circle x² + y² = 1. The quantity $\arccos(u)$ is the length of the counter-clockwise arc from (1,0) to $(u,\sqrt{1-u^2})$ on the circle x² + y² = 1.

The functions arcsine and arccosine are related by the identity

$$\arcsin(u) + \arccos(u) = \pi/2.$$

By analogy, there are functions arctangent (arctan), arccotangent (arccot), arcsecant (arcsec) and arccosecant (arccsc). They are defined as follows:

arctan:

$\arctan:(-\infty,\infty)\rightarrow(-\pi/2,\pi/2)$ by the rule

$$\arctan(u) = v\;\;{\rm if}\;\;\tan(v) = u\;\;{\rm and}\;\;v\in(-\pi/2,\pi/2)$$

Note that the range of arctangent is the range of arcsine without the endpoints. On most calculators you will not find a key sequence for arccotangent. It is approximated by relying on the identity ${\rm arccot}(u) + \arctan(u) = \pi/2$ or the identity ${\rm arccot}(u) = \arctan(1/u)$ for $x\neq 0$.

arccot:

${\rm arccot}:(-\infty,\infty)\rightarrow(0,\pi)$ by the rule

$${\rm arccot}(u) = v\;\;{\rm if}\;\;\cot(v) = u\;\;{\rm and}\;\;v\in(0,\pi)$$

Note that the range of arccotangent is the range of arccosine without the endpoints.

arcsec:

${\rm arcsec}:(-\infty,-1]\cup[1,\infty)\rightarrow[0,\pi/2)\cup(\pi/2,\pi]$ by the rule

$${\rm arcsec}(u) = v\;\;{\rm if}\;\;\sec(v) = u\;\;{\rm and}\;\;v\in[0,\pi/2)\cup(\pi/2, \pi].$$

Note that the range of arcsecant is the range of arccosine without the point $\pi/2$. In fact, since $\sec(x) = 1/\cos(x)$ we have ${\rm arcsec}(u) = \arccos(1/u)$. Generally there is no key sequence for arcsecant on a calulator, and this identity must be used to approximate its values.

arccsc:

${\rm arcsec}:(-\infty,-1]\cup[1,\infty)\rightarrow[-\pi/2,0)\cup(0,\pi/2]$ by the rule

$${\rm arccsc}(u) = v\;\;{\rm if}\;\;\csc(v) = u\;\;{\rm and}\;\;v\in[\pi/2,0)\cup(0, \pi/2].$$

Note that the range of arccosecant is the range of arcsine without the point . In fact, since $\csc(x) = 1/\sin(x)$ we have ${\rm arccsc}(u) = \arcsin(1/u)$. Generally there is no key sequence for arccosecant on a calculator, and this identity must be used to approximate its values.