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}
  • In addition to above, \csc\theta=\frac{1}{\sin\theta},\sec\theta=\frac{1}{\cos\theta},\cot\theta=\frac{1}{\tan\theta}
  • Pythagoras' formula: p^2+b^2=h^2 which leads to the trigonometric identity: \sin^2\theta+\cos^2\theta=1 and also \tan^2\theta+1=\sec^2\theta
  • 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}