Euclidean Trigonometry#
Revised
22 Mar 2023
A radian as the natural angular unit is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
In other words, an angle is equal to the ratio of the length of the arc produced on the circle to the circle’s radius.
\( \boxed{ \begin{aligned} \csc\theta&\overset{\text{def}}{=}\frac{1}{\sin\theta} \\ \sec\theta&\overset{\text{def}}{=}\frac{1}{\cos\theta} \\ \cot\theta&\overset{\text{def}}{=}\frac{\cos\theta}{\sin\theta} \end{aligned} } \)
Parity#
\( \boxed{ \begin{aligned} \sin(-\theta)&=-\sin\theta &&\text{odd function} \\ \cos(-\theta)&=\cos\theta &&\text{even function} \\ \tan(-\theta)&=-\tan\theta &&\text{odd function} \\ \csc(-\theta)&=-\csc\theta &&\text{odd function} \\ \sec(-\theta)&=\sec\theta &&\text{even function} \\ \cot(-\theta)&=-\cot\theta &&\text{odd function} \end{aligned} } \)
Pythagorean Identity#
\( \boxed{ \begin{aligned} 1=\sin^2\theta+\cos^2\theta \end{aligned} } \)
\( \boxed{ \begin{aligned} \csc^2\theta&=1+\cot^2\theta \impliedby \frac{1}{\sin^2\theta}=\frac{\sin^2\theta+\cos^2\theta}{\sin^2\theta} \\ \sec^2\theta&=\tan^2\theta+1 \impliedby \frac{1}{\cos^2\theta}=\frac{\sin^2\theta+\cos^2\theta}{\cos^2\theta} \end{aligned} } \)
Angle Sum & Difference Identities#
\( \boxed{ \begin{aligned} \sin(\alpha\pm\beta)&=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta \\ \cos(\alpha\pm\beta)&=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta \end{aligned} } \)
Derivation from Euler’s Theorem#
\( \begin{aligned} e^{i(\alpha+\beta)} &=\cos(\alpha+\beta)+i\sin(\alpha+\beta) &&\text{by Euler's Theorem} \\ e^{i(\alpha+\beta)} &=e^{i\alpha}e^{i\beta} \\ &=(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta) &&\text{by Euler's Theorem} \\ &=\cos\alpha\cos\beta+i\cos\alpha\sin\beta+i\sin\alpha\cos\beta+i^2\sin\alpha\sin\beta \\ &=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta) \\ &\implies \\ \cos(\alpha+\beta)+i\sin(\alpha+\beta) &=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta) \\ &\implies \\ \sin(\alpha+\beta)&=\sin\alpha\cos\beta+\cos\alpha\sin\beta \\ \cos(\alpha+\beta)&=\cos\alpha\cos\beta-\sin\alpha\sin\beta \end{aligned} \)
Derivation of the difference identities from the sum identities#
\( \begin{aligned} \sin(\alpha-\beta) =\sin(\alpha+(-\beta)) =\sin\alpha\cos(-\beta)+\cos\alpha\sin(-\beta) =\sin\alpha\cos\beta-\cos\alpha\sin\beta \\ \cos(\alpha-\beta) =\cos(\alpha+(-\beta)) =\cos\alpha\cos(-\beta)-\sin\alpha\sin(-\beta) =\cos\alpha\cos\beta+\sin\alpha\sin\beta \end{aligned} \)
Double Angle Identities#
\( \boxed{ \begin{aligned} \sin(2\theta)&=2\sin\theta\cos\theta \\ \cos(2\theta)&=\cos^2\theta-\sin^2\theta \end{aligned} } \)
Derivation from Euler’s Theorem#
\( \begin{aligned} e^{i(2\theta)}&=\cos(2\theta)+i\sin(2\theta) &&\text{by Euler's Theorem} \\ e^{i(2\theta)} &=(e^{i\theta})^2 \\ &=(\cos\theta+i\sin\theta)^2 &&\text{by Euler's Theorem} \\ &=\cos^2\theta+2i\cos\theta\sin\theta+i^2\sin^2\theta \\ &=(\cos^2\theta-\sin^2\theta)+i(2\cos\theta\sin\theta) \\ &\implies \\ \cos(2\theta)+i\sin(2\theta)&=(\cos^2\theta-\sin^2\theta)+i(2\cos\theta\sin\theta) \\ &\implies \\ \sin(2\theta)&=2\cos\theta\sin\theta \\ \cos(2\theta)&=\cos^2\theta-\sin^2\theta \end{aligned} \)
Half Angle Identities#
\( \boxed{ \begin{aligned} \sin^2\theta&=\frac{1-\cos(2\theta)}{2} \\ \cos^2\theta&=\frac{1+\cos(2\theta)}{2} \end{aligned} } \)
Derivation from double angle identities#
\( \begin{aligned} &\cos(2\theta) =\cos^2\theta-\sin^2\theta =(1-\sin^2\theta)-\sin^2\theta =1-2\sin^2\theta \iff \sin^2\theta=\frac{1-\cos(2\theta)}{2} \\ &\cos(2\theta) =\cos^2\theta-\sin^2\theta =\cos^2\theta-(1-\cos^2\theta) =2\cos^2\theta-1 \iff \cos^2\theta=\frac{1+\cos(2\theta)}{2} \end{aligned} \)
\( \begin{aligned} \theta=\frac{\alpha}{2} \implies \sin^2\left(\frac{\alpha}{2}\right)=\frac{1-\cos\alpha}{2} \iff \sin\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1-\cos\alpha}{2}} \\ \theta=\frac{\alpha}{2} \implies \cos^2\left(\frac{\alpha}{2}\right)=\frac{1+\cos\alpha}{2} \iff \cos\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1+\cos\alpha}{2}} \end{aligned} \)
Triple Angle Identities#
\( \boxed{ \begin{aligned} \sin(3\theta)&=3\sin\theta-4\sin^3\theta \\ \cos(3\theta)&=4\cos^3\theta-3\cos\theta \end{aligned} } \)
Derivation from Euler’s Theorem#
\( \begin{aligned} e^{i(3\theta)}&=\cos(3\theta)+i\sin(3\theta) &&\text{by Euler's Theorem} \\ e^{i(3\theta)} &=(e^{i\theta})^3 \\ &=(\cos\theta+i\sin\theta)^3 &&\text{by Euler's Theorem} \\ &=(\cos\theta+i\sin\theta)^2(\cos\theta+i\sin\theta) \\ &=(\cos^2\theta+2i\cos\theta\sin\theta+i^2\sin^2\theta)(\cos\theta+i\sin\theta) \\ &=\cos^3\theta+i\cos^2\theta\sin\theta+2i\cos^2\theta\sin\theta+2i^2\cos\theta\sin^2\theta+i^2\cos\theta\sin^2\theta+i^3\sin^3\theta \\ &=\cos^3\theta+3i\cos^2\theta\sin\theta+3i^2\cos\theta\sin^2\theta+i^3\sin^3\theta \\ &=(\cos^3\theta-3\cos\theta\sin^2\theta)+i(3\cos^2\theta\sin\theta-\sin^3\theta) \\ &=[\cos^3\theta-3\cos\theta(1-\cos^2\theta)]+i[3(1-\sin^2\theta)\sin\theta-\sin^3\theta] \\ &=(\cos^3\theta-3\cos\theta+3\cos^3\theta)+i(3\sin\theta-3\sin^3\theta-\sin^3\theta) \\ &=(4\cos^3\theta-3\cos\theta)+i(3\sin\theta-4\sin^3\theta) \\ &\implies \\ \cos(3\theta)+i\sin(3\theta) &=(4\cos^3\theta-3\cos\theta)+i(3\sin\theta-4\sin^3\theta) \\ &\implies \\ \sin(3\theta)&=3\sin\theta-4\sin^3\theta \\ \cos(3\theta)&=4\cos^3\theta-3\cos\theta \end{aligned} \)
Inverse Trigonometric Functions#
\( \boxed{ \begin{aligned} y&=\sin^{-1}x \impliedby \sin y=x &&\text{arcsin} \\ y&=\cos^{-1}x \impliedby \cos y=x &&\text{arccos} \\ y&=\tan^{-1}x \impliedby \tan y=x &&\text{arctan} \\ y&=\csc^{-1}x \impliedby \csc y=x &&\text{arccsc} \\ y&=\sec^{-1}x \impliedby \sec y=x &&\text{arcsec} \\ y&=\cot^{-1}x \impliedby \cot y=x &&\text{arccot} \\ \end{aligned} } \)
Hyperbolic Trigonometric Functions#
\( \boxed{ \begin{aligned} \sinh&\overset{\text{def}}{=}\frac{e^x-e^{-x}}{2} &&\text{hyperbolic sin} \\ \cosh&\overset{\text{def}}{=}\frac{e^x+e^{-x}}{2} &&\text{hyperbolic cos} \\ \tanh&\overset{\text{def}}{=}\frac{\sinh}{\cosh} =\frac{e^x-e^{-x}}{e^x+e^{-x}} &&\text{hyperbolic tan} \\ \text{csch}&\overset{\text{def}}{=}\frac{1}{\sinh} =\frac{2}{e^x-e^{-x}} &&\text{hyperbolic csc} \\ \text{sech}&\overset{\text{def}}{=}\frac{1}{\cosh} =\frac{2}{e^x+e^{-x}} &&\text{hyperbolic sec} \\ \text{coth}&\overset{\text{def}}{=}\frac{\cosh}{\sinh} =\frac{e^x+e^{-x}}{e^x-e^{-x}} &&\text{hyperbolic cot} \end{aligned} } \)
Terms#
[W] trigonometric identities
[W] Chord
[W] Cosine
[W] Exsecant
[W] Hyperbolic Functions
[W] Inverse Trigonometric Functions
[W] Law of Cosines
[W] Law of Sines
[W] Secant Line
[W] Sine
[W] Trigonometric Functions
[W] Trigonometric Identities, proof
[W] Unit Circle
[W] Versine