## MA101.13 Trigonometrical functions

Series definitions for the sin and cosine functions are:

$\sin{x} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$
$\cos{x} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$

These converge $\forall x \in \mathbb{R}$.

If we differentiate these term by term we can see that:

$\frac{d(\sin{x})}{dx} = \cos{x}$
$\frac{d(\cos{x})}{dx} = -\sin{x}$

Many other properties can be deduced from these power series.

The graph of $y = \sin{x}$ is as shown:

We can see that sin is not an injection (domain of $\mathbb{R}$) and so there is no inverse. However the function $f:[-\frac{\pi}{2},\frac{\pi}{2}]\to[-1,1]$, or $f:x\to \sin{x}$ [called the cut down sine], has the graph:

and this is an injection, and has an inverse function $f^{-1}$ with domain $latex[-1,1]$ and range $[-\frac{\pi}{2},\frac{\pi}{2}]$. $f^{-1}$ is the unique real number (angle) between $[-\frac{\pi}{2},\frac{\pi}{2}]$ whose sine is x. The $f^{-1}$ is symbolised by $\sin^{-1}{x}$ or $\arcsin{x}$.

The graph of $\arcsin{x}$ is a reflection of $y=\sin{x}$ (cut down) in the line $y=x$.

Knowing $y=\arcsin{x}$ is true, you may deduce $x=\sin{x}$ is true.

Example: $\frac{\pi}{4}=arcsin(\frac{1}{\sqrt{2}}) \Rightarrow sin)\frac{\pi}{4} = \frac{1}{\sqrt{2}}$.

Knowing $x = sin(y)$ is true, you may not deduce $y = arcsin(x)$.

Example: $\frac{1}{\sqrt{2}}=sin(\frac{3\pi}{4}) \nRightarrow \frac{3\pi}{4} = arcsin(\frac{1}{\sqrt{2}})$.

##### Theorem

$\frac{d(arcsin(x))}{dx} = \frac{1}{\sqrt{1-x^2}}$

###### Proof

\begin{aligned} \mbox{Let: } y &= arcsin(x) \\ \mbox{then } x &= sin(y) \\ \frac{dx}{dy} &= cos(y) \\ \frac{dy}{dx} &= \frac{1}{cos(y)} \\ &= \frac{1}{\sqrt{1-sin^2(y)}} \\ &= \frac{1}{\sqrt{1-x^2}} \end{aligned}

We take the positive square root because for $[-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}]$ the $\cos{y} \geq 0$

Similarly we define the ‘so called’ cut down cosine – this is cosine but with the domain $[0,\pi]$, and the cut down tangent with domain $(-\frac{\pi}{2}, \frac{\pi}{2})$. The inverses of these functions are arccos and arctan.