Problem Set - Trigonometry

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 February 2010

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 $\sin x$?
	Using angles, what is $\cos x$?
	Using angles, show that $\sin \left(-x\right)=-\sin x$.
	Using angles, show that $\cos \left(-x\right)=\cos x$.
	Using angles, show that ${\sin }^{2}x+{\cos }^{2}x=1$.
	Using angles, show that $\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$.
	Using angles, show that $\cos \left(x+y\right)=\cos x\cos y-\sin x\sin y$.

Computing trigonometric functions

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

Trigonometric function identities

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

Fun trigonometric functions

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

Inverse trigonometric function identities

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

Basic derivatives

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

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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