--- ## Fractions >[!quote] >--- $\large\text{Nested fractions}$ > >--- >$\Large\begin{aligned}{\color{Black}\Biggl[}\frac{\left(\frac{a}{c}\right)}{b}=\color{LightCyan}\frac{a}{bc}\color{LightGrey}{\color{Black}\Biggl]}\space\space\space\space\space\space{\color{Black}\Biggl[}\frac{a}{\left(\frac{b}{c}\right)}=\color{LightCyan}\frac{ac}{b}\color{LightGrey}{\color{Black}\Biggl]}\space\space\space\space\space\space{\color{Black}\Biggl[}\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\color{LightCyan}\frac{ad}{bc}\color{LightGrey}{\color{Black}\Biggl]}\end{aligned}$ >$\newline$ > >--- >$\Large\text{Fraction identities}$ > >--- >$\Large\begin{aligned}{\color{Black}\Biggl[}\frac{a-b}{c-d}=\color{LightCyan}\frac{b-a}{d-c}\color{LightGrey}{\color{Black}\Biggl]}\space\space\space\space\space\space\space\space\space\space\space\space{\color{Black}\Biggl[}\frac{a}{b}+\frac{c}{d}=\color{LightCyan}\frac{ad-bc}{bd}\color{LightGrey}{\color{Black}\Biggl]}\end{aligned}$ --- ## Exponents >[!quote] >--- >$\Large\text{Exponential identities}$ > >--- >$\huge\begin{aligned}\color{black}\boxed{\color{LightGrey}x^0=\color{LightCyan}1}\\\color{black}\boxed{\color{LightGrey}1^a=\color{LightCyan}1}\\\color{black}\boxed{\color{LightGrey}\Big(x^a\Big)^b=\color{LightCyan}x^{a\cdot b}}\\\color{black}\boxed{\color{LightGrey}\left(\frac{x}{y}\right)^a=\color{LightCyan}\frac{(x)^a}{(y)^a}}\\\color{black}\boxed{\color{LightGrey}\frac{x^a}{x^b}=\color{LightCyan}x^{a-b}}\end{aligned}\space\space\space\space\space\space\space\begin{align*}&\color{black}\boxed{\color{LightGrey}x^1=\color{LightCyan}a}\\&\color{black}\boxed{\color{LightGrey}a^{-1}=\color{LightCyan}\frac{1}{a}}\\&\color{black}\boxed{\color{LightGrey}x^a\cdot x^b=\color{LightCyan}x^{a+b}}\\&\color{black}\boxed{\color{LightGrey}\Big(xy\Big)^a=\color{LightCyan}x^a\cdot y^a}\\&\color{black}\boxed{\color{LightGrey}\frac{x^a}{x^b}=\color{LightCyan}\frac{1}{x^{b-a}}}\end{align*}$ --- ## Radicals >[!quote] >--- >$\Large\text{Radical identities}$ > >--- >$\Large\begin{aligned}\sqrt[a]{x^b}&=\color{LightCyan}x^{\frac{b}{a}}\\\\\Big(\sqrt[a]{x}\Big)^b&=\color{LightCyan}\sqrt[a]{x^b}\\\\\sqrt[a]{xy}&=\color{LightCyan}\sqrt[a]{x}\cdot\sqrt[a]{y}\end{aligned}\space\space\space\space\space\space\space\space\space\space\space\space\begin{aligned}\sqrt[a]{\frac{x}{y}}&=\color{LightCyan}\frac{\sqrt[a]{x}}{\sqrt[a]{y}}\\\\\large\sqrt[a]{\sqrt[b]{x}}&=\color{lightCyan}\sqrt[a\cdot b]{x}\\\\\big(\sqrt[a]{x}\big)^a&=\color{LightCyan}x\end{aligned}$ --- ## Log and natural log >[!quote] >--- >$\Large\text{Logarithm definition}$ > >--- >$\huge\begin{aligned}\color{black}\boxed{\color{LightGrey}\log_ab=x\longrightarrow\color{LightCyan}a^x=b}\\\end{aligned}$ >$\Large\log_a\big(1\big)=\color{LightCyan}0\color{LightGrey}\space\longrightarrow \big(a^0=1\big)$ >$\Large\log_a\big(a\big)=\color{LightCyan}1\color{LightGrey}\space\longrightarrow \big(a^1=a\big)$ >$\Large\log_a\big(0\big)=\color{LightCyan}\textit{undefined}$ >$\Large\log_a\big(-b\big)=\color{lightCyan}\textit{undefined}$ >$\newline$ > >--- >$\Large\text{Logarithm identities}$ > >--- >$\Large\begin{aligned}\log_ca+\log_cb&=\color{LightCyan}\log\big(a\cdot b\big)\\\\\log_ca-\log_cb&=\color{LightCyan}\log\left(\frac{a}{b}\right)\\\\log_c\big(\sqrt[b]{a}\big)&=\color{LightCyan}\frac{\log_ca}{b}\\\\a\log_c\big(x\big)+b\log_c\big(y\big)&=\color{LightCyan}\log_c\big(x^ay^b\big)\end{aligned}\space\space\space\space\space\space\space\space\begin{aligned}b\cdot\log_ca&=\color{LightCyan}\log_c\Big(a^b\Big)\\\\x^{\log_ay}&=y^{\log_ax}\\\\\log_a\big(x\big)&=\color{LightCyan}\frac{\log_bx}{\log_ba}\\\\\log_ab&=\color{LightCyan}\frac{1}{\log_ba}\end{aligned}\space\space\space\space\space\space\space\space$ >$\newline$ --- ## Trigonometry ##### *Pythagorean identity and basic definitions* >[!quote] >--- >$\Large\text{The Pythagorean identities}$ > >--- >$\huge\color{black}\boxed{\color{LightGrey}\begin{aligned}\sin^2\theta\space{\color{LightCyan}+}\space\cos^2\theta&=\color{LightCyan}1\\\color{LightGrey}\sec^2\theta\space{\color{LightCyan}-}\space\tan^2\theta\color{lightCyan}&=\color{LightCyan}1\end{aligned}}$ >$\newline$ > >--- >$\Large\text{Defining relationships}$ > >--- >$\Large\begin{aligned}\space\space\space\space\sec\theta&=\color{LightCyan}\frac{1}{\cos\theta}\\\\\csc\theta&=\color{LightCyan}\frac{1}{\sin\theta}\end{aligned}\space\space\space\space\space\space\space\space\space\space\space\space\space\space\begin{aligned}\tan\theta&=\color{LightCyan}\frac{\sin\theta}{\cos\theta}\\\\\cot\theta&=\color{LightCyan}\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}\end{aligned}$ ##### *Angle identities* >[!quote] >--- >$\Large\text{Double and tripple angle identities}$ > >--- >$\Large\begin{aligned}\\\cos2\theta&=\color{LightCyan}\cos^2\theta-\sin^2\theta\\&=\color{LightCyan}2\cos^2\theta-1\\&=\color{LightCyan}1-2\sin^2\theta\\\\\sin2\theta&=\color{LightCyan}2\sin\theta\cos\theta\\\\\tan2\theta&=\color{LightCyan}\frac{2\tan\theta}{1-\tan^2\theta}\end{aligned}\space\space\space\space\space\space\space\space\space\space\begin{aligned}\sin3\theta&=\color{LightCyan}3\sin\theta-4\sin^3\theta\\\\\cos3\theta&=\color{LightCyan}4\cos^3\theta-3\cos\theta\\\\\tan3\theta&=\color{LightCyan}\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}\\\\\end{aligned}$ >[!quote] >$\large\text{Inverse double angle identities}$ > >--- >$\Large\begin{aligned}\;\;\;\;\;\;\;\;\;\;\cos^2(x)&=\color{LightCyan}\frac{1}{2}+\frac{1}{2}\cos(2x)\\\\\sin^2(x)&=\color{LightCyan}\frac{1}{2}-\frac{1}{2}\cos(2x)\\\\\tan^2(x)&=\color{LightCyan}-2\tan(x)\cot(2x)+1\end{aligned}$ ##### *Sum and product identities* >[!quote] >--- >$\Large\text{Sum and difference identities}$ > >--- >$\Large\begin{aligned}\sin(a\pm b)&=\color{LightCyan}\sin(a)\cos(b)\pm\sin(b)\cos(a)\\\\\cos(a\pm b)&=\color{LightCyan}\cos(a)\cos(b)\pm\sin(a)\sin(b)\end{aligned}$ >$\large\begin{aligned}\\\tan(a+b)=\color{LightCyan}\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}\color{LightGrey}\space\space\space\space\tan(a-b)=\color{LightCyan}\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}\end{aligned}$ ##### *Complements* >[!quote] >--- >$\Large\text{Complement identities}$ > >--- >$\Large\begin{aligned}\sin\theta&=\color{LightCyan}\sin\left(\frac{\pi}{2}-\theta\right)\\\\-\cos\theta&=\color{LightCyan}\cos\left(\frac{\pi}{2}-\theta\right)\\\\-\tan\theta&=\color{LightCyan}\tan\left(\frac{\pi}{2}-\theta\right)\end{aligned}\space\space\space\space\space\space\space\space\space\space\space\space\space\begin{aligned}-\cot\theta&=\color{LightCyan}\cot\left(\frac{\pi}{2}-\theta\right)\\\\\csc\theta&=\color{LightCyan}\csc\left(\frac{\pi}{2}-\theta\right)\\\\-\sec\theta&=\color{LightCyan}\sec\left(\frac{\pi}{2}-\theta\right)\end{aligned}$ >$\newline$ >--- >$\Large\text{Negative angle identites}$ > >--- >$\newline$ >$\Large\sin(-\theta)=\color{LightCyan}-\sin\theta\;\;\;|\;\;\color{LightGrey}\cos(-\theta)=\color{LightCyan}\cos\theta\;\;\;|\;\;\color{LightGrey}\tan(-\theta)=\color{LightCyan}-\tan\theta$ >$\newline$ >--- >$\Large\text{Periodicity rules}$ > >--- >$\newline$ >$\Large\sin(\theta+2\pi)=\color{LightCyan}\sin\theta\;\;|\;\color{LightGrey}\cos(\theta+2\pi)=\color{LightCyan}\cos\theta\;\;|\;\color{LightGrey}\tan(\theta+\pi)=\color{LightCyan}\tan\theta$ ##### *Nested trig terms* >[!quote] >##### *Nested trigonometric functions* >--- >$\text{Nested inverse sin terms}$ >$\Huge\boxed{\normalsize\;\;\begin{aligned}\;\\\cos\Big(\sin^{-1}(x)\Big)&=\color{LightCyan}\sqrt{1-x^2}\\\\\tan\Big(\sin^{-1}(x)\Big)&=\color{LightCyan}\frac{x}{\sqrt{1-x^2}}\\\;\end{aligned}\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}\;\\\sec\Big(\sin^{-1}(x)\Big)&=\color{LightCyan}\frac{1}{\sqrt{1-x^2}}\\\\\csc\Big(\sin^{-1}(x)\Big)&=\color{LightCyan}\frac{1}{x}\\\;\end{aligned}\;\;}$ > >--- >$\text{Nested inverse cosine terms}$ >$\Huge\boxed{\normalsize\;\;\begin{aligned}\;\\\sin\Big(\cos^{-1}(x)\Big)&=\color{LightCyan}\sqrt{1-x^2}\\\\\tan\Big(\cos^{-1}(x)\Big)&=\color{LightCyan}\frac{\sqrt{1-x^2}}{x}\\\;\end{aligned}\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}\;\\\sec\Big(\cos^{-1}(x)\Big)&=\color{LightCyan}\frac{1}{x}\\\\\csc\Big(\cos^{-1}(x)\Big)&=\color{LightCyan}\frac{1}{\sqrt{1-x^2}}\\\;\end{aligned}}$ > >--- >$\text{Nested inverse tangent terms}$ >$\Huge\boxed{\normalsize\;\;\begin{aligned}\;\\\sin\Big(\tan^{-1}(x)\Big)&=\color{LightCyan}\frac{x}{\sqrt{1+x^2}}\\\\\cos\Big(\tan^{-1}(x)\Big)&=\color{LightCyan}\frac{1}{\sqrt{1+x^2}}\\\;\end{aligned}\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}\;\\\sec\Big(\tan^{-1}(x)\Big)&=\color{LightCyan}\sqrt{1+x^2}\\\\\csc\Big(\tan^{-1}(x)\Big)&=\color{LightCyan}\frac{\sqrt{1+x^2}}{x}\\\;\end{aligned}\;\;}$ --- #subpage