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