MathematicsTrigonometry

Trigonometery

For any right triangle with hypotenuse hh, an acute angle α\alpha, side length oo opposite from α\alpha, and side length aa adjacent to α\alpha, the following terms are defined:

sine α= sin⁡α = ohcosine α= cos⁡α = ahtangent α= tan⁡α = oa=sin⁡αcosαcotangent α= cot⁡α =ctn α = ao=1tan⁡α=cos⁡αsin⁡αsecant α=sec⁡α=ha=1cos⁡αcosecant α=csc⁡α=ho=1sin⁡αexsecant α=exsec α=sec⁡α−1versine α=vers α=1−cos⁡αcoversine α=covers α=1−sin⁡αhaversine α=hav α=vers α2\begin{align} \text{sine } \alpha &= \sin{\alpha} = \frac{o}{h} \\ \text{cosine } \alpha &= \cos{\alpha} = \frac{a}{h} \\ \text{tangent } \alpha &= \tan{\alpha} = \frac{o}{a} = \frac{\sin{\alpha}}{cos{\alpha}} \\ \text{cotangent } \alpha &= \cot{\alpha} = \text{ctn } \alpha = \frac{a}{o} = \frac{1}{\tan{\alpha}} = \frac{\cos{\alpha}}{\sin{\alpha}} \\ \text{secant } \alpha &= \sec{\alpha} = \frac{h}{a} = \frac{1}{\cos{\alpha}} \\ \text{cosecant } \alpha &= \csc{\alpha} = \frac{h}{o} = \frac{1}{\sin{\alpha}} \\ \text{exsecant } \alpha &= \text{exsec } \alpha = \sec{\alpha} - 1 \\ \text{versine } \alpha &= \text{vers } \alpha = 1 - \cos{\alpha} \\ \text{coversine } \alpha &= \text{covers } \alpha = 1 - \sin{\alpha} \\ \text{haversine } \alpha &= \text{hav } \alpha = \frac{\text{vers } \alpha}{2} \\ \end{align}

also defined are the following…

hyperbolic sine of x= sinh⁡x = ex−e−x2hyperbolic cosine of x= cosh⁡x = ex+e−x2hyperbolic tangent of x= tanh⁡x = sinh⁡xcosh⁡x=ex−e−xex+e−xcsch x = 1sinh⁡xsech x = 1cosh⁡xcoth x = 1tanh⁡x\begin{align} \text{hyperbolic sine of } x &= \sinh{x} = \frac{\mathrm{e}^x - \mathrm{e}^{-x}}{2} \\ \text{hyperbolic cosine of } x &= \cosh{x} = \frac{\mathrm{e}^x + \mathrm{e}^{-x}}{2} \\ \text{hyperbolic tangent of } x &= \tanh{x} = \frac{\sinh{x}}{\cosh{x}} = \frac{\mathrm{e}^x - \mathrm{e}^{-x}}{\mathrm{e}^x + \mathrm{e}^{-x}} \\ \text{csch } x &= \frac{1}{\sinh{x}} \\ \text{sech } x &= \frac{1}{\cosh{x}} \\ \text{coth } x &= \frac{1}{\tanh{x}} \\ \end{align}

Identities

Pythagorean Identities

sin⁡2α+cos⁡2α=11+tan⁡2α=sec⁡2α1+cot⁡2α=csc⁡2α\begin{align} \sin^2{\alpha} + \cos^2{\alpha} &= 1 \\ 1 + \tan^2{\alpha} &= \sec^2{\alpha} \\ 1 + \cot^2{\alpha} &= \csc^2{\alpha} \\ \end{align}

Half Angle Identities

sin⁡α2=±1−cos⁡α2 (negative if α2 is in quadrant III or IV)cos⁡α2=±1+cos⁡α2 (negative if α2 is in quadrant II or III)tan⁡α2=±1−cos⁡α1+cos⁡α (negative if α2 is in quadrant II or IV)\begin{align} \sin{\frac{\alpha}{2}} &= \pm \sqrt{\frac{1 - \cos{\alpha}}{2}} \text{ (negative if } \frac{\alpha}{2} \text{ is in quadrant III or IV)}\\ \cos{\frac{\alpha}{2}} &= \pm \sqrt{\frac{1 + \cos{\alpha}}{2}} \text{ (negative if } \frac{\alpha}{2} \text{ is in quadrant II or III)}\\ \tan{\frac{\alpha}{2}} &= \pm \sqrt{\frac{1 - \cos{\alpha}}{1 + \cos{\alpha}}} \text{ (negative if } \frac{\alpha}{2} \text{ is in quadrant II or IV)}\\ \end{align}

Double-Angle Identities

sin⁡2α=2sin⁡αcos⁡αcos⁡2α=2cos⁡2α−1=1−2sin⁡2α=cos⁡2α−sin⁡2αtan⁡2α=2tan⁡α1−tan⁡2α\begin{align} \sin{2\alpha} &= 2\sin{\alpha}\cos{\alpha}\\ \cos{2\alpha} &= 2\cos^2{\alpha} - 1 = 1 - 2\sin^2{\alpha} = \cos^2{\alpha} - \sin^2{\alpha}\\ \tan{2\alpha} &= \frac{2\tan{\alpha}}{1 - \tan^2{\alpha}}\\ \end{align}

n-Angle Identities

sin⁡3α=3sin⁡α−4sin⁡3αcos⁡3α=4cos⁡3α−3cos⁡αsin⁡nα=2sin⁡((n−1)α)cos⁡α−sin⁡(n−2)αcos⁡nα=2cos⁡((n−1)α)cos⁡α−cos⁡(n−2)α\begin{align} \sin{3\alpha} &= 3\sin{\alpha} - 4\sin^3{\alpha}\\ \cos{3\alpha} &= 4\cos^3{\alpha} - 3\cos{\alpha}\\ \sin{n\alpha} &= 2\sin\big(\left(n-1\right)\alpha\big) \cos{\alpha} - \sin(n-2)\alpha\\ \cos{n\alpha} &= 2\cos\big(\left(n-1\right)\alpha\big) \cos{\alpha} - \cos\left(n-2\right)\alpha\\ \end{align}

Two-Angle Identities

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡βcos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡βtan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡βsin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡βcos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡βtan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\begin{align} \sin\left(\alpha + \beta\right) &= \sin{\alpha}\cos{\beta} + \cos{\alpha}\sin{\beta}\\ \cos\left(\alpha + \beta\right) &= \cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta}\\ \tan\left(\alpha + \beta\right) &= \frac{\tan{\alpha} + \tan{\beta}}{1 - \tan{\alpha}\tan{\beta}}\\ \sin\left(\alpha - \beta\right) &= \sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}\\ \cos\left(\alpha - \beta\right) &= \cos{\alpha}\cos{\beta} + \sin{\alpha}\sin{\beta}\\ \tan\left(\alpha - \beta\right) &= \frac{\tan{\alpha} - \tan{\beta}}{1 + \tan{\alpha}\tan{\beta}}\\ \end{align}

Sum and Difference Identities

sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2cos⁡α+cos⁡β=2cos⁡α+β2sin⁡α−β2cos⁡α−cos⁡β=−2cos⁡α+β2sin⁡α−β2tan⁡α+tan⁡β=sin⁡(α+β)cos⁡αcos⁡βcot⁡α+cot⁡β=sin⁡(α+β)sin⁡αsin⁡βtan⁡α−tan⁡β=sin⁡(α−β)cos⁡αcos⁡βcot⁡α−cot⁡β=−sin⁡(α−β)sin⁡αsin⁡βsin⁡2α−sin⁡2β=sin⁡(α+β)sin⁡(α−β)cos⁡2α−cos⁡2β=−sin⁡(α+β)sin⁡(α−β)cos⁡2α−sin⁡2β=cos⁡(α+β)cos⁡(α−β)\begin{align} \sin{\alpha} + \sin{\beta} &= 2\sin{\frac{\alpha + \beta}{2}}\cos{\frac{\alpha - \beta}{2}}\\ \sin{\alpha} - \sin{\beta} &= 2\cos{\frac{\alpha + \beta}{2}}\sin{\frac{\alpha - \beta}{2}}\\ \cos{\alpha} + \cos{\beta} &= 2\cos{\frac{\alpha + \beta}{2}}\sin{\frac{\alpha - \beta}{2}}\\ \cos{\alpha} - \cos{\beta} &= -2\cos{\frac{\alpha + \beta}{2}}\sin{\frac{\alpha - \beta}{2}}\\ \tan{\alpha} + \tan{\beta} &= \frac{\sin\left(\alpha + \beta\right)}{\cos{\alpha}\cos{\beta}}\\ \cot{\alpha} + \cot{\beta} &= \frac{\sin\left(\alpha + \beta\right)}{\sin{\alpha}\sin{\beta}}\\ \tan{\alpha} - \tan{\beta} &= \frac{\sin\left(\alpha - \beta\right)}{\cos{\alpha}\cos{\beta}}\\ \cot{\alpha} - \cot{\beta} &= -\frac{\sin\left(\alpha - \beta\right)}{\sin{\alpha}\sin{\beta}}\\ \sin^2{\alpha} - \sin^2{\beta} &= \sin\left(\alpha + \beta\right) \sin\left(\alpha - \beta\right)\\ \cos^2{\alpha} - \cos^2{\beta} &= -\sin\left(\alpha + \beta\right) \sin\left(\alpha - \beta\right)\\ \cos^2{\alpha} - \sin^2{\beta} &= \cos\left(\alpha + \beta\right) \cos\left(\alpha - \beta\right)\\ \end{align}

Power Identities

sin⁡αsin⁡β=cos⁡(α−β)−cos⁡(α+β)2cos⁡αcos⁡β=cos⁡(α−β)+cos⁡(α+β)2sin⁡αcos⁡β=sin⁡(α+β)+sin⁡(α−β)2cos⁡αsin⁡β=sin⁡(α+β)−sin⁡(α−β)2tan⁡αcot⁡α=sin⁡αcsc⁡α=cos⁡αsec⁡α=1sin⁡2α=1−cos⁡2α2cos⁡2α=1+cos⁡2α2sin⁡3α=3sin⁡α−sin⁡3α4cos⁡3α=3cos⁡α+cos⁡3α4sin⁡4α=3−4cos⁡2α+cos⁡4α8cos⁡4α=3+4cos⁡2α+cos⁡4α8sin⁡5α=10sin⁡α−5sin⁡3α+sin⁡5α16cos⁡5α=10cos⁡α+5cos⁡3α+cos⁡5α16\begin{align} \sin{\alpha}\sin{\beta} &= \frac{\cos\left(\alpha - \beta\right) - \cos\left(\alpha + \beta\right)}{2}\\ \cos{\alpha}\cos{\beta} &= \frac{\cos\left(\alpha - \beta\right) + \cos\left(\alpha + \beta\right)}{2}\\ \sin{\alpha}\cos{\beta} &= \frac{\sin\left(\alpha + \beta\right) + \sin\left(\alpha - \beta\right)}{2}\\ \cos{\alpha}\sin{\beta} &= \frac{\sin\left(\alpha + \beta\right) - \sin\left(\alpha - \beta\right)}{2}\\ \tan{\alpha}\cot{\alpha} &= \sin{\alpha}\csc{\alpha} = \cos{\alpha}\sec{\alpha} = 1\\ \sin^2{\alpha} &= \frac{1 - \cos{2\alpha}}{2}\\ \cos^2{\alpha} &= \frac{1 + \cos{2\alpha}}{2}\\ \sin^3{\alpha} &= \frac{3\sin{\alpha} - \sin{3\alpha}}{4}\\ \cos^3{\alpha} &= \frac{3\cos{\alpha} + \cos{3\alpha}}{4}\\ \sin^4{\alpha} &= \frac{3 - 4\cos{2\alpha} + \cos{4\alpha}}{8}\\ \cos^4{\alpha} &= \frac{3 + 4\cos{2\alpha} + \cos{4\alpha}}{8}\\ \sin^5{\alpha} &= \frac{10\sin{\alpha} - 5\sin{3\alpha} + \sin{5\alpha}}{16}\\ \cos^5{\alpha} &= \frac{10\cos{\alpha} + 5\cos{3\alpha} + \cos{5\alpha}}{16}\\ \end{align}

OBLIQUE TRIANGLES

(no right angle, angles A,B,C are opposite of legs a,b,c)

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}

Law of Cosines
a2=b2+c2−2bccos⁡Ab2=a2+c2−2accos⁡Bc2=a2+b2−2abcos⁡Ccos⁡C=a2+b2−c22ab\begin{align} a^2 &= b^2 + c^2 - 2bc\cos{A}\\ b^2 &= a^2 + c^2 - 2ac\cos{B}\\ c^2 &= a^2 + b^2 - 2ab\cos{C}\\ \cos{C} &= \frac{a^2 +b^2 -c^2}{2ab}\\ \end{align}
Law of Tangents

a−ba+b=tan⁡a−b2tan⁡a+b2\frac{a-b}{a+b} = \frac{\tan\frac{a-b}{2}}{\tan\frac{a+b}{2}}

Projection Formulas
a=bcos⁡C+ccos⁡Bb=ccos⁡A+acos⁡Cc=acos⁡B+bcos⁡A\begin{align} a &= b\cos{C} + c\cos{B}\\ b &= c\cos{A} + a\cos{C}\\ c &= a\cos{B} + b\cos{A}\\ \end{align}
Mollweide’s Check Formulas
a−bc=sin⁡A−B2cos⁡C2a+bc=cos⁡A−B2sin⁡C2\begin{align} \frac{a-b}{c} &= \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}}\\ \frac{a+b}{c} &= \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}}\\ \end{align}
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