Analys-och-Linj-r-algibra/Trigonometri.md

• Radian:
  • Def: It is the SI unit for measuring angles (in the plane).
  • Def: 1 radian is defined as the angle subtended at the center by a circular arc of length equal to the radius
  • Def: A general angle is measured in radians as the ration of the length an associated circular arc and the corresponding radius. That is $\theta=\frac{s}{r}\text{rad}$
  • Def: Usually "$rad$" is omitted.
  • Ex: \begin{align}180^\circ=\pi\text{ rad}\\\frac{\pi}{3}\text{ rad}=30^\circ\\\frac{\pi}{4}\text{ rad}=45^\circ\\\frac{\pi}{3}\text{ rad}=60^\circ\\\frac{\pi}{2}\text{ rad}=90^\circ\\2\pi\text{ rad}=360^\circ\end{align}
• The right angled triangle
  • Def: *The trigonometric functions: *\begin{align}\sin\theta=\frac{\text{perpendicular}}{\text{hypotenuse}}\\\cos\theta=\frac{\text{base}}{\text{hypotenuse}}\\\tan\theta=\frac{\text{perpendicular}}{\text{base}}\end{align}
  • Dominains and ranges:
    • D_{\sin}=\mathbb{R}\;\;R_{\sin}=[-1,1]
    • D_{\cos}=\mathbb{R}\;\;R_{\cos}=[-1,1]
    • D_{\tan}=\mathbb{R}\setminus\{n\pi+\frac{\pi}{2}:n\in\mathbb{Z}\}\;\;R_{\tan}=(-\infty,\infty)
• Useful relations
  • \sin(-\theta)=-\sin(\text{odd}),\cos(-\theta)=\cos\theta(\text{even})
  • Periodicity: \sin(\theta+2n\pi)=\sin\theta,\cos(\theta+2n\pi)=\cos\theta,\tan(\theta+n\pi)=\tan\theta
  • Complementary angles: \sin(\frac{\pi}{2}-\theta)=\cos\theta,\cos(\frac{\pi}{2}-\theta)=\sin\theta
  • Sift by \pi: \sin(\theta\pm\pi)=-\sin\theta,\cos(\theta\pm\pi=-\cos\theta
  • Sum of angles: \sin(\theta+\phi)=\sin\theta\times\cos\phi+\cos\theta\times\sin\phi,\cos(\theta+\phi)=\cos\theta\times\cos\phi-\sin\theta\times\sin\phi,\tan(\theta+\phi)=\frac{\tan\theta+\tan\phi}{1-\tan\theta\times\tan\phi}
  • Double angle: \sin(2\theta)=2\sin\theta\cos\theta,\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta},\cos(2\theta)=\cos^2-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta
  • Half angle: 2\sin^2\frac{\theta}{2}=1-\cos\theta,2\cos^2\frac{\theta}{2}=1+\cos\theta
• Solving trigonometric equations
  • \sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.
  • \cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.
  • \tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}
  • Ex: Solve \sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}
• Inverse trigonometric function
  • Def: f(x)=\sin(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right]. Then f is strictly increasing on D_f and hence inverible. The fuction \arcsin is defined as $\arcsin(x)=f^{-1}(x)\text{ on }D_{arcsin}=R_f=[-1,1]$
  • Similarly: For g(x)=\cos(x),x\in\left[0,\pi\right] (which is strictly decreasing) and h(x)=\tan(x),x\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right] (which is strictly increasing), the function \arccos and \arctan are defined as $\begin{align}\arccos(x)=g^{-1}(x)\text{ on }D_{\arccos}=\left[-1,1\right]\\\arctan(x)=h^{-1}(x)\text{ on }D_{\arctan}=\mathbb{R}\end{align}$
  • Note: That the tanges R_{\arcsin}=R_{\arctan}=\left[\frac{-\pi}{2},\frac{\pi}{2}\right] whereas $R_{\arccos}=\left[0,\pi\right]$
• Properties
  • Def: \begin{align}\sin(\arcsin(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arcsin(sin(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\\cos(\arccos(x))=x\forall{x}\in\left[-1,1\right]\text{ | }\arccos(\cos(x))=x\text{ if }x\in\left[0,\pi\right]\\\tan(\arctan(x))=x\forall{x}\in\mathbb{R}\text{ | }\arctan(\tan(x))=x\text{ if }x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\end{align}
  • Complementary angles: \arcsin(x)+\arccos(x)=\frac{pi}{2},\;\arctan(x)+\arccos(x)=\frac{\pi}{2}
  • Negatives: \arcsin and \arctan are odd functions. $\begin{align}\arcsin(-x)=-\arcsin(x)\\\arccos(-x)=\pi-\arccos(x)\\\arctan(-x)=-\arctan(x)\end{align}$