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Sinus Cosinus Tangens Arctan

Definition

Trigonometry studies relationships involving lengths and angles of triangles.

Unit Circle

Sin, Cos, Tan

sin2(α)+cos2(α)=1\sin^2(\alpha) + \cos^2(\alpha) = 1

with angle α\alpha

Explanation

In the unit circle, the sinus and cosinus at a given angle form an orthogonal triangle with the edges a,b,ca,b,c. The length of the hypotenuse cc corresponds to the radius rr, which equals 1 (unit circle). Thus, we can apply the law of Pythagoras a2+b2=c2a^2 + b^2 = c^2 with a=sin(α)a = \sin(\alpha), b=cos(α)b = \cos(\alpha), and c=1c = 1.

sin(α)=br\sin(\alpha) = \frac{b}{r}

with angle α\alpha, radius rr

Symmetry: sin(α)=sin(α)\sin(\alpha) = - \sin(\alpha)

cos(α)=ar\cos(\alpha) = \frac{a}{r}

with angle α\alpha, radius rr

Symmetry: cos(α)=cos(α)\cos(\alpha) = \cos(-\alpha)

tan(α)=cos(α)sin(α)=ba\tan(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{b}{a}

with angle α\alpha


Properties Equation
Symmetry sin(x)=sin(x)\sin(-x)=-\sin(x) cos(x)=cos(x)\cos (-x) = \cos (x)
Complex eix=cos(x)+isin(x)e^{\text{i}x}=\cos(x)+\text{i}\sin(x) eix=sin(x)icos(x)e^{-\text{i}x}=\sin(x)-\text{i}\cos(x)
xx
$\scriptstyle{ \alpha }$
00
$\scriptstyle{0^\circ}$
π/6\pi / 6
$\scriptstyle{30^\circ}$
π/4\pi / 4
$\scriptstyle{45^\circ}$
π/3\pi / 3
$\scriptstyle{60^\circ}$
12π\frac{1}{2}\pi
$\scriptstyle{90^\circ}$
π\pi
$\scriptstyle{180^\circ}$
112π1\frac{1}{2}\pi
$\scriptstyle{270^\circ}$
2π2 \pi
$\scriptstyle{360^\circ}$
sin\sin 00 12\frac{1}{2} 12\frac{1}{\sqrt{2}} 32\frac{\sqrt 3}{2} 11 00 1-1 00
cos\cos 11 32\frac{\sqrt 3}{2} 12\frac{1}{\sqrt 2} 12\frac{1}{2} 00 1-1 00 11
tan\tan 00 33\frac{\sqrt{3}}{3} 11 3\sqrt{3} ±\pm \infty 00 \mp \infty 00
Addition Integrals
cos(xπ2)=sinx\cos (x - \frac{\pi}{2}) = \sin x xcos(x)dx=cos(x)+xsin(x)\int x \cos(x) \,\text{d}x = \cos(x) + x \sin(x)
sin(x+π2)=cosx\sin (x + \frac{\pi}{2}) = \cos x xsin(x)dx=sin(x)xcos(x)\int x \sin(x) \,\text{d}x = \sin(x) - x \cos(x)
sin2x=2sinxcosx\sin 2x = 2 \sin x \cos x sin2(x)dx=12(xsin(x)cos(x))\int \sin^2(x) \,\text{d}x = \frac12 \bigl(x - \sin(x)\cos(x) \bigr)
cos2x=2cos2x1\cos 2x = 2\cos^2 x - 1 cos2(x)dx=12(x+sin(x)cos(x))\int \cos^2(x) \,\text{d}x = \frac12 \bigl(x + \sin(x)\cos(x) \bigr)
sin(x)=tan(x)cos(x)\sin(x) = \tan(x)\cos(x) cos(x)sin(x)=12cos2(x)\int \cos(x)\sin(x) = -\frac12 \cos^2(x)

Hyperboles sinh, cosh, tanh

cosh(x)+sinh(x)=ex\cosh (x) + \sinh (x) = e^{x}
cosh2(x)sinh2(x)=1\cosh^2 (x) - \sinh^2 (x) = 1

sinh(x)=12(exex)=isin(ix)\sinh(x) = \frac{1}{2}(e^x -e^{-x}) = - \text{i}\, \sin(\text{i}x)

cosh(x)=12(ex+ex)=cos(ix)\cosh(x) = \frac{1}{2}(e^x +e^{-x}) = \cos(\text{i}x)

tanh(x)=sinhxcoshx=exexex+ex=e2x1e2x+1=12e2x+1\tanh(x) =\frac {\sinh x}{\cosh x} = {\frac {\mathrm {e} ^{x}-\mathrm {e} ^{-x}}{\mathrm {e} ^{x}+\mathrm {e} ^{-x}}}={\frac {\mathrm {e} ^{2x}-1}{\mathrm {e} ^{2x}+1}}=1-{\frac {2}{\mathrm {e} ^{2x}+1}}

coth(x)=coshxsinhx=ex+exexex=e2x+1e2x1=1+2e2x1\coth(x) ={\frac {\cosh x}{\sinh x}}={\frac {\mathrm {e} ^{x}+\mathrm {e} ^{-x}}{\mathrm {e} ^{x}-\mathrm {e} ^{-x}}}={\frac {\mathrm {e} ^{2x}+1}{\mathrm {e} ^{2x}-1}}=1+{\frac {2}{\mathrm {e} ^{2x}-1}}


Inverse

arcsinh(x):=ln(x+x2+1)\mathrm{arcsinh}(x) := \ln\left(x+\sqrt{x^2+1}\right)

arccosh(x):=ln(x+x21)\mathrm{arccosh}(x) := \ln\left(x+\sqrt{x^2-1}\right)

artanh(x)=12ln1+x1x\mathrm{artanh}(x) = \frac{1}{2} \ln {\frac {1+x}{1-x}}

arcoth(x)=12lnx+1x1\mathrm{arcoth}(x) = \frac{1}{2} \ln {\frac {x+1}{x-1}}


Cardinal Sinus

si(x)=sin(x)x\mathrm{si}(x) = \frac{\sin(x)}{x}

normalized: $$\sinc(x) = \frac{\sin(\pi x)}{\pi x}$$