Problem Set - Trigonometry

## Angles

 What is $\pi$ and where did it come from? Explain how to measure angles in radians, in degrees and how to convert between them. What is the connection between measuring angles in radians and measuring distance? What is the circumference of a circle of radius $r$? How do you know? What is the length of an arc of angle $\theta$ on the boundary of a circle of radius $r$? How do you know? What is the area of a circle of radius $r$? How do you know? What is the area of a sector of angle $\theta$ in a circle of radius $r$? How do you know? Using angles, what is $\mathrm{sin}x$? Using angles, what is $\mathrm{cos}x$? Using angles, show that $\mathrm{sin}\left(-x\right)=-\mathrm{sin}x$. Using angles, show that $\mathrm{cos}\left(-x\right)=\mathrm{cos}x$. Using angles, show that ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$. Using angles, show that $\mathrm{sin}\left(x+y\right)=\mathrm{sin}x\mathrm{cos}y+\mathrm{cos}x\mathrm{sin}y$. Using angles, show that $\mathrm{cos}\left(x+y\right)=\mathrm{cos}x\mathrm{cos}y-\mathrm{sin}x\mathrm{sin}y$.

## Computing trigonometric functions

 Explain how to derive $\mathrm{sin}\frac{\pi }{6},\mathrm{cos}\frac{\pi }{6},\mathrm{tan}\frac{\pi }{6},\mathrm{cot}\frac{\pi }{6},\mathrm{sec}\frac{\pi }{6}$ and $\mathrm{csc}\frac{\pi }{6}$ in radical form. Explain how to derive $\mathrm{sin}\frac{\pi }{3},\mathrm{cos}\frac{\pi }{3},\mathrm{tan}\frac{\pi }{3},\mathrm{cot}\frac{\pi }{3},\mathrm{sec}\frac{\pi }{3}$ and $\mathrm{csc}\frac{\pi }{3}$ in radical form. Explain how to derive $\mathrm{sin}\frac{\pi }{4},\mathrm{cos}\frac{\pi }{4},\mathrm{tan}\frac{\pi }{4},\mathrm{cot}\frac{\pi }{4},\mathrm{sec}\frac{\pi }{4}$ and $\mathrm{csc}\frac{\pi }{4}$ in radical form. Explain how to derive $\mathrm{sin}\frac{\pi }{2},\mathrm{cos}\frac{\pi }{2},\mathrm{tan}\frac{\pi }{2},\mathrm{cot}\frac{\pi }{2},\mathrm{sec}\frac{\pi }{2}$ and $\mathrm{csc}\frac{\pi }{2}$ in radical form. Explain how to derive $\mathrm{sin}0,\mathrm{cos}0,\mathrm{tan}0,\mathrm{cot}0,\mathrm{sec}0$ and $\mathrm{csc}0$ in radical form. Explain how to derive $\mathrm{sin}\frac{3\pi }{4},\mathrm{cos}\frac{3\pi }{4},\mathrm{tan}\frac{3\pi }{4},\mathrm{cot}\frac{3\pi }{4},\mathrm{sec}\frac{3\pi }{4}$ and $\mathrm{csc}\frac{3\pi }{4}$ in radical form. Explain how to derive $\mathrm{sin}\frac{-2\pi }{3},\mathrm{cos}\frac{-2\pi }{3},\mathrm{tan}\frac{-2\pi }{3},\mathrm{cot}\frac{-2\pi }{3},\mathrm{sec}\frac{-2\pi }{3}$ and $\mathrm{csc}\frac{-2\pi }{3}$ in radical form. Compute $\mathrm{sin}\frac{\pi }{6}+\mathrm{cos}\frac{\pi }{6}$ in radical form. Compute $\left(\mathrm{sin}\frac{\pi }{6}\right)\left(\mathrm{cos}\frac{\pi }{6}\right)$ in radical form. Compute $\left(\mathrm{tan}\frac{\pi }{6}\right)\left(\mathrm{cot}\frac{\pi }{6}\right)$ in radical form.

## Trigonometric function identities

 Verify the identity $\frac{\mathrm{sec}A-1}{\mathrm{sec}A+1}+\frac{\mathrm{cos}A-1}{\mathrm{cos}A+1}=0$. Verify the identity $\mathrm{sin}V\left(1+{\mathrm{cot}}^{2}V\right)=\mathrm{csc}V$. Verify the identity $\frac{\mathrm{sin}\left(\pi /2-w\right)}{\mathrm{cos}\left(\pi /2-w\right)}=\mathrm{cot}w$. Verify the identity $\mathrm{sec}\left(\pi /2-z\right)=\frac{1}{\mathrm{sin}z}$. Verify the identity $1+{\mathrm{tan}}^{2}\left(\pi /2-x\right)=\frac{1}{{\mathrm{cos}}^{2}\left(\pi /2-x\right)}$. Verify the identity $\frac{\mathrm{sin}A}{\mathrm{csc}A}+\frac{\mathrm{cos}A}{\mathrm{sec}A}=1$. Verify the identity $\frac{\mathrm{sec}B}{\mathrm{cos}B}-\frac{\mathrm{tan}B}{\mathrm{cot}B}=0$. Verify the identity $\frac{1}{{\mathrm{csc}}^{2}w}+{\mathrm{sec}}^{2}w+\frac{1}{{\mathrm{sec}}^{2}w}=2+\frac{{\mathrm{sec}}^{2}w}{{\mathrm{csc}}^{2}w}$. Verify the identity ${\mathrm{sec}}^{4}V-{\mathrm{sec}}^{2}V=\frac{1}{{\mathrm{cot}}^{4}V}+\frac{1}{{\mathrm{cot}}^{2}V}$. Verify the identity ${\mathrm{sin}}^{4}x+{\mathrm{cos}}^{2}x={\mathrm{cos}}^{4}x+{\mathrm{sin}}^{2}x$. Verify the identity $\mathrm{tan}3\alpha =\frac{3\mathrm{tan}\alpha -{\mathrm{tan}}^{3}\alpha }{1-3{\mathrm{tan}}^{2}\alpha }$. Verify the identity $\mathrm{cot}\left(\alpha /2\right)=\frac{\mathrm{sin}\alpha }{1-\mathrm{cos}\alpha }$. Verify the identity $\mathrm{cos}\left(\pi /6-x\right)+\mathrm{cos}\left(\pi /6+x\right)=\sqrt{3}\mathrm{cos}x$. Verify the identity $\mathrm{sin}\left(\alpha +\beta \right)\mathrm{sin}\left(\alpha -\beta \right)={\mathrm{sin}}^{2}\alpha -{\mathrm{sin}}^{2}\beta$. Verify the identity $\mathrm{sin}\left(\pi /3-x\right)+\mathrm{sin}\left(\pi /3+x\right)=\sqrt{3}\mathrm{cos}x$. Verify the identity $\mathrm{cos}\left(\pi /4-x\right)-\mathrm{cos}\left(\pi /4+x\right)=\sqrt{2}\mathrm{sin}x$. Verify the identity $2\mathrm{sin}\alpha \mathrm{cos}\beta =\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)$. Verify the identity $2\mathrm{sin}\alpha \mathrm{sin}\beta =\mathrm{cos}\left(\alpha -\beta \right)-\mathrm{cos}\left(\alpha +\beta \right)$.

## Fun trigonometric functions

 Verify the identity $\mathrm{cos}2\theta =2\mathrm{sin}\left(\pi /4+\theta \right)\mathrm{sin}\left(\pi /4-\theta \right)$ Verify the identity $\frac{\mathrm{sin}2A}{2}=\frac{\mathrm{tan}A}{1+{\mathrm{tan}}^{2}A}$. Verify the identity $\mathrm{cot}\left(x/2\right)=\frac{1+\mathrm{cos}x}{\mathrm{sin}x}$. Verify the identity $\mathrm{sin}2B\left(\mathrm{cot}B+\mathrm{tan}B\right)=2$. Verify the identity $\frac{1-{\mathrm{tan}}^{2}\theta }{1+{\mathrm{tan}}^{2}\theta }=\mathrm{cos}2\theta$. Verify the identity $1+\mathrm{cos}2A=\frac{2}{1+{\mathrm{tan}}^{2}A}$. Verify the identity $\mathrm{tan}2x\mathrm{tan}x+2=\frac{\mathrm{tan}2x}{\mathrm{tan}x}$. Verify the identity $\mathrm{csc}A\mathrm{sec}A=2\mathrm{csc}2A$. Verify the identity $\mathrm{cot}x=\frac{\mathrm{sin}2x}{1-\mathrm{cos}2x}$. Verify the identity $1-\mathrm{sin}A={\left(\mathrm{sin}\frac{A}{2}-\mathrm{cos}\frac{A}{2}\right)}^{2}$. Verify the identity ${\mathrm{cos}}^{4}A=\frac{2\mathrm{cos}2A+{\mathrm{cos}}^{2}2A+1}{4}$. Verify the identity $\frac{\mathrm{sin}A+\mathrm{sin}B}{\mathrm{sin}A-\mathrm{cos}A}=\frac{\mathrm{tan}\left(\frac{A+B}{2}\right)}{\mathrm{tan}\left(\frac{A-B}{2}\right)}$. Verify the identity $\frac{\mathrm{sin}\alpha +\mathrm{sin}3\alpha }{\mathrm{cos}\alpha +\mathrm{cos}3\alpha }=\mathrm{tan}2\alpha$. Verify the identity $\frac{\mathrm{cos}2A}{1+\mathrm{sin}2A}=\frac{\mathrm{cot}A-1}{\mathrm{cot}A+1}$. Verify the identity $\frac{\mathrm{cos}A+\mathrm{sin}A}{\mathrm{cos}A-\mathrm{sin}A}=\frac{1+\mathrm{sin}2A}{\mathrm{cos}2A}$. Verify the identity $\mathrm{cot}\alpha -\mathrm{cot}\beta =\frac{\mathrm{sin}\left(\beta -\alpha \right)}{\mathrm{sin}\alpha \mathrm{sin}\beta }$. Verify the identity $\mathrm{tan}\theta \mathrm{csc}\theta \mathrm{cos}\theta =1$. Verify the identity ${\mathrm{cos}}^{2}\theta =\frac{{\mathrm{cot}}^{2}\theta }{1+{\mathrm{cot}}^{2}\theta }$. Verify the identity $\frac{1-\mathrm{sin}A}{1+\mathrm{sin}A}={\left(\mathrm{sec}A-\mathrm{tan}A\right)}^{2}$. Verify the identity ${\left(\mathrm{tan}A-\mathrm{cot}A\right)}^{2}+4={\mathrm{sec}}^{2}A+{\mathrm{csc}}^{2}A$. Verify the identity $\mathrm{cos}B\mathrm{cos}\left(A+B\right)+\mathrm{sin}B\mathrm{sin}\left(A+B\right)=\mathrm{cos}B$. Verify the identity $\frac{\mathrm{tan}A-\mathrm{sin}A}{\mathrm{sec}A}=\frac{{\mathrm{sin}}^{3}A}{1+\mathrm{cos}A}$. Verify the identity $\frac{2{\mathrm{tan}}^{2}A}{1+{\mathrm{tan}}^{2}A}=1-\mathrm{cos}2A$. Verify the identity $\mathrm{tan}2A=\mathrm{tan}A+\frac{\mathrm{tan}A}{\mathrm{cos}2A}$. Verify the identity $\mathrm{sin}2A=\frac{2\mathrm{tan}A}{1+{\mathrm{tan}}^{2}A}$. Verify the identity $\frac{4\mathrm{sin}A}{1-{\mathrm{sin}}^{2}A}=\frac{1+\mathrm{sin}A}{1-\mathrm{sin}A}-\frac{1-\mathrm{sin}A}{1+\mathrm{sin}A}$. Verify the identity $\mathrm{tan}A+\mathrm{sin}A=\frac{\mathrm{csc}A+\mathrm{cot}A}{\mathrm{csc}A\mathrm{cot}A}$.

## Inverse trigonometric function identities

 Verify the identity $\mathrm{cos}\left({\mathrm{tan}}^{-1}x\right)=\frac{1}{\sqrt{1+{x}^{2}}}$. Verify the identity $\mathrm{sin}\left({\mathrm{tan}}^{-1}x\right)=\frac{x}{\sqrt{1+{x}^{2}}}$. Verify the identity $\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)=\sqrt{1-{x}^{2}}$. Verify the identity $\mathrm{tan}\left({\mathrm{cos}}^{-1}x\right)=\frac{\sqrt{1-{x}^{2}}}{x}$. Verify the identity $\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)=\sqrt{1-{x}^{2}}$. Verify the identity $\mathrm{tan}\left({\mathrm{cot}}^{-1}x\right)=\frac{1}{x}$. Verify the identity $\mathrm{cot}\left({\mathrm{cot}}^{-1}2\right)=2$. Verify the identity $\mathrm{sin}\left({\mathrm{cot}}^{-1}x\right)=\frac{1}{\sqrt{1+{x}^{2}}}$. Verify the identity $\mathrm{cos}\left({\mathrm{cot}}^{-1}x\right)=\frac{x}{\sqrt{1+{x}^{2}}}$. Verify the identity ${\mathrm{sin}}^{-1}\left(-x\right)=-{\mathrm{sin}}^{-1}x$. Verify the identity ${\mathrm{tan}}^{-1}\left(-x\right)=-{\mathrm{tan}}^{-1}x$. Verify the identity ${\mathrm{tan}}^{-1}x={\mathrm{cot}}^{-1}\left(1/x\right)$. Verify the identity ${\mathrm{tan}}^{-1}x={\mathrm{sin}}^{-1}\left(\frac{x}{\sqrt{1+{x}^{2}}}\right)$. Verify the identity ${\mathrm{sin}}^{-1}\left(\frac{x}{\sqrt{1+{x}^{2}}}\right)={\mathrm{cos}}^{-1}\left(\frac{x}{\sqrt{1+{x}^{2}}}\right)$.

## Basic derivatives

 What is $\frac{d}{dx}$? Explain why $\frac{d1}{dx}=0$. Explain why $\frac{da}{dx}=0$ if $a$ is a number. Explain why $\frac{dx}{dx}=1$. Explain why $\frac{d{x}^{2}}{dx}=2x$. Explain why $\frac{d{x}^{3}}{dx}=3{x}^{2}$. Explain why $\frac{d{x}^{-1}}{dx}=-{x}^{-2}$. Explain why $\frac{d{x}^{-2}}{dx}=-2{x}^{-3}$. Explain why $\frac{d{x}^{-3}}{dx}=-3{x}^{-4}$. Explain why $\frac{d{\left(3{x}^{2}+2x\right)}^{-1}}{dx}=\frac{-\left(6x+2\right)}{{\left(3{x}^{2}+2x\right)}^{2}}$. Explain why $\frac{d{x}^{1/2}}{dx}=\frac{1}{2}{x}^{-1/2}$. Explain why $\frac{d{x}^{1/3}}{dx}=\frac{1}{3}{x}^{-2/3}$. Explain why $\frac{d{x}^{3/5}}{dx}=\frac{3}{5}{x}^{-2/5}$. Explain why $\frac{d{x}^{n}}{dx}=n{x}^{n-1}$, for all positive integers $n$. Explain why $\frac{d{x}^{n}}{dx}=n{x}^{n-1}$, for $n=0$. Explain why $\frac{d{x}^{n}}{dx}=n{x}^{n-1}$, for all negative integers $n$. Explain why $\frac{d{x}^{m/n}}{dx}=\frac{m}{n}{x}^{\left(m/n\right)-1}$ for all integers $m$ and $n$, with $n\ne 0$.