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The study of Angles and of the angular relationships of planar and 3-D figures is known as trigonometry. The trigonometric functions (also called the Circular Functions) comprising trigonometry are the Cosecant $\csc x$, Cosine $\cos x$, Cotangent $\cot x$, Secant $\sec x$, Sine $\sin x$, and Tangent $\tan x$. The inverses of these functions are denoted $\csc^{-1} x$, $\cos^{-1} x$, $\cot^{-1} x$, $\sec^{-1} x$, $\sin^{-1} x$, and $\tan^{-1} x$. Note that the $f^{-1}$ Notation here means Inverse Function, not $f$ to the $-1$ Power.


The trigonometric functions are most simply defined using the Unit Circle. Let $\theta$ be an Angle measured counterclockwise from the x-Axis along an Arc of the Circle. Then $\cos\theta$ is the horizontal coordinate of the Arc endpoint, and $\sin\theta$ is the vertical component. The Ratio $\sin\theta/\cos\theta$ is defined as $\tan\theta$. As a result of this definition, the trigonometric functions are periodic with period $2\pi$, so

\mathop{\rm func}(2\pi n+\theta) = \mathop{\rm func}(\theta),
\end{displaymath} (1)

where $n$ is an Integer and func is a trigonometric function.

From the Pythagorean Theorem,

\sin^2\theta+\cos^2\theta = 1.
\end{displaymath} (2)

Therefore, it is also true that
\tan^2\theta+1 = \sec^2\theta
\end{displaymath} (3)

1+\cot^2\theta = \csc^2\theta.
\end{displaymath} (4)

The trigonometric functions can be defined algebraically in terms of Complex Exponentials (i.e., using the Euler Formula) as
$\displaystyle \sin z$ $\textstyle \equiv$ $\displaystyle {e^{iz}-e^{-iz}\over 2i}$ (5)
$\displaystyle \csc z$ $\textstyle \equiv$ $\displaystyle {1\over \sin z} = {2i\over e^{iz}-e^{-iz}}$ (6)
$\displaystyle \cos z$ $\textstyle \equiv$ $\displaystyle {e^{iz}+e^{-iz}\over 2}$ (7)
$\displaystyle \sec z$ $\textstyle \equiv$ $\displaystyle {1\over \cos z}= {2\over e^{iz}+e^{-iz}}$ (8)
$\displaystyle \tan z$ $\textstyle \equiv$ $\displaystyle {\sin z\over \cos z}= {e^{iz}-e^{-iz}\over i(e^{iz}+e^{-iz})}$ (9)
$\displaystyle \cot z$ $\textstyle \equiv$ $\displaystyle {1\over \tan z}= {i(e^{iz}+e^{-iz})\over e^{iz}-e^{-iz}}
= {i(1+e^{-2iz})\over 1-e^{-2iz}}.$ (10)

Osborne's Rule gives a prescription for converting trigonometric identities to analogous identities for Hyperbolic Functions.

The Angles $n\pi/m$ (with $m,n$ integers) for which the trigonometric function may be expressed in terms of finite root extraction of real numbers are limited to values of $m$ which are precisely those which produce constructible Polygons. Gauß showed these to be of the form

m=2^k p_1p_2\cdots p_s,
\end{displaymath} (11)

where $k$ is an Integer $\geq 0$ and the $p_i$ are distinct Fermat Primes. The first few values are $m=1$, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, ... (Sloane's A003401). Although formulas for trigonometric functions may be found analytically for other $m$ as well, the expressions involve Roots of Complex Numbers obtained by solving a Cubic, Quartic, or higher order equation. The cases $m=7$ and $m=9$ involve the Cubic Equation and Quartic Equation, respectively. A partial table of the analytic values of Sine, Cosine, and Tangent for arguments $\pi/m$ is given below. Derivations of these formulas appear in the following entries.

$x$ (°) $x$ (rad) $\sin x$ $\cos x$ $\tan x$
0.0 0 0 1 0
15.0 ${\textstyle{1\over 12}}\pi$ ${\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2})$ ${\textstyle{1\over 4}}(\sqrt{6}+\sqrt{2})$ $2-\sqrt{3}$
18.0 ${\textstyle{1\over 10}}\pi$ ${\textstyle{1\over 4}}(\sqrt{5}\,-1)$ ${\textstyle{1\over 4}}\sqrt{10+2\sqrt{5}}$ ${\textstyle{1\over 5}}\sqrt{25-10\sqrt{5}}$
22.5 ${\textstyle{1\over 8}}\pi$ ${\textstyle{1\over 2}}\sqrt{2-\sqrt{2}}$ ${\textstyle{1\over 2}}\sqrt{2+\sqrt{2}}$ $\sqrt{2}\,-1$
30.0 ${\textstyle{1\over 6}}\pi$ ${\textstyle{1\over 2}}$ ${\textstyle{1\over 2}}\sqrt{3}$ ${\textstyle{1\over 3}}\sqrt{3}$
36.0 ${\textstyle{1\over 5}}\pi$ ${\textstyle{1\over 4}}\sqrt{10-2\sqrt{5}}$ ${\textstyle{1\over 4}}(1+\sqrt{5}\,)$ $\sqrt{5-2\sqrt{5}}$
45.0 ${\textstyle{1\over 4}}\pi$ ${\textstyle{1\over 2}}\sqrt{2}$ ${\textstyle{1\over 2}}\sqrt{2}$ 1
60.0 ${\textstyle{1\over 3}}\pi$ ${\textstyle{1\over 2}}\sqrt{3}$ ${\textstyle{1\over 2}}$ $\sqrt{3}$
90.0 ${\textstyle{1\over 2}}\pi$ 1 0 $\infty$
180.0 $\pi$ 0 $-1$ 0

The Inverse Trigonometric Functions are generally defined on the following domains.

Function Domain
$\sin^{-1} x$ $-{\textstyle{1\over 2}}\pi \leq y \leq {\textstyle{1\over 2}}\pi $
$\cos^{-1} x$ $0 \leq y \leq \pi$
$\tan^{-1} x$ $-{\textstyle{1\over 2}}\pi < y < {\textstyle{1\over 2}}\pi $
$\csc^{-1} x$ $0 \leq y \leq {\textstyle{1\over 2}}\pi $ or $\pi \leq y \leq {\textstyle{3\pi\over 2}}$
$\sec^{-1} x$ $0 \leq y \leq \pi$
$\cot^{-1} x$ $0 \leq y \leq {\textstyle{1\over 2}}\pi $ or $-\pi \leq y \leq -{\textstyle{1\over 2}}\pi $

Inverse-forward identities are

\tan^{-1}(\cot x) = {\textstyle{1\over 2}}\pi - x
\end{displaymath} (12)

\sin^{-1}(\cos x) = {\textstyle{1\over 2}}\pi - x
\end{displaymath} (13)

\sec^{-1}(\csc x) = {\textstyle{1\over 2}}\pi - x,
\end{displaymath} (14)

and forward-inverse identities are
\cos(\sin^{-1} x) = \sqrt{1-x^2}
\end{displaymath} (15)

\cos(\tan^{-1} x) = {1\over \sqrt{1+x^2}}
\end{displaymath} (16)

\sin(\cos^{-1} x) = \sqrt{1-x^2}

\sin(\tan^{-1} x) = {x\over \sqrt{1+x^2}}
\end{displaymath} (17)

\tan(\cos^{-1} x) = {\sqrt{1-x^2}\over x}
\end{displaymath} (18)

\tan(\sin^{-1} x) = {x\over \sqrt{1-x^2}}.
\end{displaymath} (19)

Inverse sum identities include

\sin^{-1} x+\cos^{-1} x = {\textstyle{1\over 2}}\pi
\end{displaymath} (20)

\tan^{-1} x+\cot^{-1} x = {\textstyle{1\over 2}}\pi
\end{displaymath} (21)

\sec^{-1} x+\csc^{-1} x = {\textstyle{1\over 2}}\pi ,
\end{displaymath} (22)

where (20) follows from
x = \sin(\sin^{-1}x) = \cos({\textstyle{1\over 2}}\pi -\sin^{-1}x).
\end{displaymath} (23)

Complex inverse identities in terms of Logarithms include

$\displaystyle \sin^{-1}(z)$ $\textstyle =$ $\displaystyle -i \ln(iz\pm \sqrt{1-z^2}\,)$ (24)
$\displaystyle \cos^{-1}(z)$ $\textstyle =$ $\displaystyle -i \ln(z\pm i\sqrt{1-z^2}\,)$ (25)
$\displaystyle \tan^{-1}(z)$ $\textstyle =$ $\displaystyle -i \ln\left({1+iz\over \sqrt{1+z^2}}\right)$ (26)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}i\ln\left({1-iz\over 1+iz}\right).$ (27)

For Imaginary arguments,

$\displaystyle \sin(iz)$ $\textstyle =$ $\displaystyle i\sinh z$ (28)
$\displaystyle \cos(iz)$ $\textstyle =$ $\displaystyle \cosh z.$ (29)

For Complex arguments,
$\displaystyle \sin(x+iy)$ $\textstyle =$ $\displaystyle \sin x\cosh y+i\cos x\sinh y$ (30)
$\displaystyle \cos(x+iy)$ $\textstyle =$ $\displaystyle \cos x\cosh y-i\sin x\sinh y.$ (31)

For the Absolute Square of Complex arguments $z=x+iy$,
$\displaystyle \vert\sin(x+iy)\vert^2$ $\textstyle =$ $\displaystyle \sin^2 x+\sinh^2 y$ (32)
$\displaystyle \vert\cos(x+iy)\vert^2$ $\textstyle =$ $\displaystyle \cos^2 x+\sinh^2 y.$ (33)

The Modulus also satisfies the curious identity
\vert\sin(x+iy)\vert=\vert\sin x+\sin(iy)\vert.
\end{displaymath} (34)

The only functions satisfying identities of this form,
\vert f(x+iy)\vert=\vert f(x)+f(iy)\vert
\end{displaymath} (35)

are $f(z)=Az$, $f(z)=A\sin(bz)$, and $f(z)=A\sinh(bz)$ (Robinson 1957).

Trigonometric product formulas can be derived using the following figure (Kung 1996).

\begin{figure}\begin{center}\BoxedEPSF{TrigSumProduct.epsf scaled 550}\end{center}\end{figure}

In the figure,

$\displaystyle \theta$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\alpha-\beta)$ (36)
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\alpha+\beta),$ (37)

$\displaystyle s$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\sin\alpha+\sin\beta)=\cos[{\textstyle{1\over 2}}(\alpha-\beta)]\sin[{\textstyle{1\over 2}}(\alpha+\beta)]$  
$\displaystyle t$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\cos\alpha+\cos\beta)=\cos[{\textstyle{1\over 2}}(\alpha-\beta)]\cos[{\textstyle{1\over 2}}(\alpha+\beta)].$  

\begin{figure}\begin{center}\BoxedEPSF{TrigDiffProduct.epsf scaled 550}\end{center}\end{figure}

With $\theta$ and $\gamma$ as previously defined, the above figure (Kung 1996) gives

$\displaystyle u$ $\textstyle =$ $\displaystyle \cos\beta-\cos\alpha=2\sin[{\textstyle{1\over 2}}(\alpha-\beta)]\sin[{\textstyle{1\over 2}}(\alpha+\beta)]$  
$\displaystyle v$ $\textstyle =$ $\displaystyle \sin\alpha-\sin\beta=2\sin[{\textstyle{1\over 2}}(\alpha-\beta)]\cos[{\textstyle{1\over 2}}(\alpha+\beta)].$  

Angle addition Formulas express trigonometric functions of sums of angles $\alpha\pm\beta$ in terms of functions of $\alpha$ and $\beta$. They can be simply derived using Complex exponentials and the Euler Formula,

$\displaystyle \sin(\alpha+\beta)$ $\textstyle =$ $\displaystyle {e^{i(\alpha+\beta)}-e^{-i(\alpha+\beta)}\over 2i} = {e^{i\alpha}e^{i\beta}-e^{-i\alpha}e^{-i\beta}\over 2i}$  
  $\textstyle =$ $\displaystyle {(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta)\over 2i}$  
  $\textstyle \phantom{=}$ $\displaystyle -{(\cos\alpha-i\sin\alpha)(\cos\beta-i\sin\beta)\over 2i}$  
  $\textstyle =$ $\displaystyle {\cos\alpha\cos\beta+i\sin\beta\cos\alpha+i\sin\alpha\cos\beta-\sin\alpha\sin\beta\over 2i}$  
  $\textstyle \phantom{=}$ $\displaystyle + {-\cos\alpha\cos\beta+i\cos\alpha\sin\beta+i\sin\alpha\cos\beta+\sin\alpha\sin\beta\over 2i}$  
  $\textstyle =$ $\displaystyle \sin \alpha\cos \beta+\sin \beta\cos \alpha$ (42)

$\displaystyle \cos(\alpha+\beta)$ $\textstyle =$ $\displaystyle {e^{i(\alpha+\beta)}+e^{-i(\alpha+\beta)}\over 2} = {e^{i\alpha}e^{i\beta}+e^{-i\alpha}e^{-i\beta}\over 2}$  
  $\textstyle =$ $\displaystyle {(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta)\over 2}$  
  $\textstyle \phantom{=}$ $\displaystyle +{(\cos\alpha-i\sin\alpha)(\cos\beta-i\sin\beta)\over 2}$  
  $\textstyle =$ $\displaystyle {\cos\alpha\cos\beta+i\cos\alpha\sin\beta+i\sin\alpha\cos\beta-\sin\alpha\sin\beta\over 2}$  
  $\textstyle \phantom{=}$ $\displaystyle +{\cos\alpha\cos\beta-i\cos\alpha\sin\beta-i\sin\alpha\cos\beta-\sin\alpha\sin\beta\over 2}$  
  $\textstyle =$ $\displaystyle \cos\alpha\cos\beta-\sin\alpha\sin\beta.$ (43)

Taking the ratio gives the tangent angle addition Formula

$\displaystyle \tan(\alpha+\beta)$ $\textstyle \equiv$ $\displaystyle {\sin(\alpha+\beta)\over\cos(\alpha+\beta)} = {\sin\alpha\cos\beta+\sin\beta\cos\alpha\over\cos\alpha\cos\beta-\sin\alpha\sin\beta}$  
  $\textstyle =$ $\displaystyle {{\sin\alpha\over\cos\alpha}+{\sin\beta\over\cos\beta}\over 1-{\s...
...cos\alpha\cos\alpha\beta}} = {\tan\alpha+\tan\beta\over 1-\tan\alpha\tan\beta}.$ (44)

The angle addition Formulas can also be derived purely algebraically without the use of Complex Numbers. Consider the following figure.


From the large Right Triangle,

$\displaystyle \sin(\alpha+\beta)$ $\textstyle =$ $\displaystyle {L\sin\beta+a\over L\cos\alpha+b}$ (45)
$\displaystyle \cos(\alpha+\beta)$ $\textstyle =$ $\displaystyle {L\cos\beta \over L\cos\alpha+b}.$ (46)

But, from the small triangle (inset at upper right),
$\displaystyle a$ $\textstyle =$ $\displaystyle {L\sin\alpha \over \cos(\alpha+\beta)}$ (47)
$\displaystyle b$ $\textstyle =$ $\displaystyle L\sin\alpha \tan(\alpha+\beta).$ (48)

Plugging $a$ and $b$ from (47) and (48) into (45) and (46) gives
$\displaystyle \sin(\alpha+\beta)$ $\textstyle =$ $\displaystyle {L\sin\beta+{L\sin\alpha\over\cos(\alpha+\beta)}\over L\cos\alpha+{L\sin\alpha\sin(\alpha+\beta)\over\cos(\alpha+\beta)}}$  
  $\textstyle =$ $\displaystyle {\sin\beta\cos(\alpha+\beta)+\sin\alpha\over \cos\alpha\cos(\alpha+\beta)+\sin\alpha\sin(\alpha+\beta)},$  

$\displaystyle \cos(\alpha+\beta)$ $\textstyle =$ $\displaystyle {L \cos\beta\over L\cos\alpha+ {L\sin\alpha\sin(\alpha+\beta)\over\cos(\alpha+\beta)}}$  
  $\textstyle =$ $\displaystyle {\cos\beta\over\cos\alpha+{\sin\alpha\sin(\alpha+\beta)\over\cos(\alpha+\beta)}}.$ (50)

Now solve (50) for $\cos(\alpha+\beta)$,
\end{displaymath} (51)

to obtain
\end{displaymath} (52)

Plugging (52) into (49) gives

$={\sin\beta \cos\beta-\sin\alpha\sin\beta\sin(\alpha+\beta)+\sin\alpha\cos\alph...
$={\sin\alpha\cos\alpha+\sin\beta\cos\beta\over\cos\alpha\cos\beta} -{\sin\alpha\sin\beta\over\cos\alpha\cos\beta}\sin(\alpha+\beta),$ (53)


\end{displaymath} (54)

\sin(\alpha+\beta)(\cos\alpha\cos\beta+\sin\alpha\sin\beta) =\sin\alpha\cos\alpha+\sin\beta\cos\beta,
\end{displaymath} (55)


$\displaystyle \sin(\alpha+\beta)$ $\textstyle =$ $\displaystyle {\sin\alpha\cos\alpha+\sin\beta\cos\beta\over\sin\alpha\sin\beta+\cos\alpha\cos\beta}$  
  $\textstyle =$ $\displaystyle {\sin\alpha\cos\alpha+\sin\beta\cos\beta\over\sin\alpha\sin\beta+...
...lpha\cos\beta+\sin\beta\cos\alpha\over\sin\alpha\cos\beta+\sin\beta\cos\alpha}.$ (56)

Multiplying out the Denominator gives

$= \sin\alpha\cos\alpha\cos^2\beta+\cos^2\alpha\sin\beta\cos\beta$
$=\sin\alpha\cos\alpha+\sin\beta\cos\beta,$ (57)


\sin(\alpha+\beta) = \sin\alpha\cos\beta+\sin\beta\cos\alpha.
\end{displaymath} (58)

Multiplying out (50),
\end{displaymath} (59)

$\displaystyle \cos(\alpha+\beta)$ $\textstyle =$ $\displaystyle {\cos\beta-\sin\alpha\sin(\alpha+\beta)\over\cos\alpha}$  
  $\textstyle =$ $\displaystyle {\cos\beta-\sin\alpha(\sin\alpha\cos\beta+\sin\beta\cos\alpha)\over\cos\alpha}$  
  $\textstyle =$ $\displaystyle {\cos\beta(1-\sin^2\alpha)-\sin\alpha\cos\alpha\sin\beta\over\cos\alpha}$  
  $\textstyle =$ $\displaystyle {\cos^2\alpha\cos\beta-\sin\alpha\cos\alpha\sin\beta\over\cos\alpha}$  
  $\textstyle =$ $\displaystyle \cos\alpha\cos\beta-\sin\alpha\sin\beta.$ (60)

Summarizing (and explicitly writing out the identities for which $\beta$ is taken to be explicitly negative),

$\displaystyle \sin(\alpha+\beta)$ $\textstyle =$ $\displaystyle \sin \alpha\cos \beta+\sin \beta\cos \alpha$ (61)
$\displaystyle \sin(\alpha-\beta)$ $\textstyle =$ $\displaystyle \sin\alpha\cos\beta-\sin\beta\cos\alpha$ (62)
$\displaystyle \cos(\alpha+\beta)$ $\textstyle =$ $\displaystyle \cos\alpha\cos\beta-\sin\alpha\sin\beta$ (63)
$\displaystyle \cos(\alpha-\beta)$ $\textstyle =$ $\displaystyle \cos\alpha\cos\beta+\sin\alpha\sin\beta$ (64)
$\displaystyle \tan(\alpha+\beta)$ $\textstyle =$ $\displaystyle {\tan\alpha+\tan\beta\over 1-\tan\alpha\tan\beta}$ (65)
$\displaystyle \tan(\alpha-\beta)$ $\textstyle =$ $\displaystyle {\tan\alpha-\tan\beta\over 1+\tan\alpha\tan\beta}.$ (66)

The sine and cosine angle addition identities can be summarized by the Matrix Equation

\left[{\matrix{\cos x & \sin x\cr -\sin x & \cos x\cr}}\righ...
...x{\cos(x+y) & \sin(x+y)\cr -\sin(x+y) & \cos(x+y)\cr}}\right].
\end{displaymath} (67)

The double angle formulas are

$\displaystyle \sin(2\alpha)$ $\textstyle =$ $\displaystyle 2\sin\alpha\cos\alpha$ (68)
$\displaystyle \cos(2\alpha)$ $\textstyle =$ $\displaystyle \cos^2\alpha-\sin^2\alpha$ (69)
  $\textstyle =$ $\displaystyle 2\cos^2\alpha-1$ (70)
  $\textstyle =$ $\displaystyle 1-2\sin^2\alpha$ (71)
$\displaystyle \tan(2\alpha)$ $\textstyle =$ $\displaystyle {2\tan\alpha\over 1-\tan^2\alpha}.$ (72)

General multiple angle formulas are

$\sin(n\alpha) = 2\sin[(n-1)\alpha]\cos\alpha-\sin[(n-2)\alpha]$ (73)
$\sin(nx) = n\cos^{n-1}x\sin x-{n(n-1)(n-2)\over 1\cdot 2\cdot 3}\cos^{n-3}x\sin^3 x+\ldots$ (74)
$\cos(n\alpha) = 2\cos[(n-1)\alpha]\cos\alpha-\cos[(n-2)\alpha]$ (75)
$\cos(nx) =\cos^n x-{n(n-1)\over 1\cdot 2} \cos^{n-2} x\sin^2 x$
$ + {n(n-1)(n-2)(n-3)\over 1\cdot 2\cdot 3\cdot 4} \cos^{n-4} x\sin^4 x-\ldots\quad$ (76)
$\tan(n\alpha) = {\tan[(n-1)\alpha]+\tan\alpha\over 1-\tan[(n-1)\alpha]\tan\alpha}.$ (77)
Therefore, any trigonometric function of a sum can be broken up into a sum of trigonometric functions with $\sin\alpha\cos\alpha$ cross terms. Particular cases for multiple angle formulas up to $n=4$ are given below.

$\displaystyle \sin(3\alpha)$ $\textstyle =$ $\displaystyle 3\sin\alpha-4\sin^3\alpha$ (78)
$\displaystyle \cos(3\alpha)$ $\textstyle =$ $\displaystyle 4\cos^3\alpha-3\cos\alpha$ (79)
$\displaystyle \tan(3\alpha)$ $\textstyle =$ $\displaystyle {3\tan\alpha-\tan^3\alpha\over 1-3\tan^2\alpha}$ (80)
$\displaystyle \sin(4\alpha)$ $\textstyle =$ $\displaystyle 4\sin\alpha\cos\alpha-8\sin^3 \alpha\cos\alpha$ (81)
$\displaystyle \cos(4\alpha)$ $\textstyle =$ $\displaystyle 8\cos^4\alpha-8\cos^2\alpha+1$ (82)
$\displaystyle \tan(4\alpha)$ $\textstyle =$ $\displaystyle {4\tan\alpha-4\tan^3\alpha\over 1-6\tan^2\alpha+\tan^4\alpha}.$ (83)

Beyer (1987, p. 139) gives formulas up to $n=6$.

Sum identities include

$\displaystyle {\tan(\alpha-\beta)\over\tan(\alpha+\beta)}$ $\textstyle =$ $\displaystyle {\sin(\alpha-\beta)\cos(\alpha+\beta)\over\cos(\alpha-\beta)\sin(\alpha+\beta)}$  
  $\textstyle =$ $\displaystyle {(\sin\alpha\cos\beta-\sin\beta\cos\alpha)(\cos\alpha\cos\beta-\s...
  $\textstyle =$ $\displaystyle {\sin\alpha\cos\alpha-\sin\beta\cos\beta\over\sin\alpha\cos\alpha+\sin\beta\cos\beta}.$ (84)

Infinite sum identities include

\sum_{k=1,\ 3,\ 5,\ldots}^\infty {e^{-kx}\sin(ky)\over k} = {1\over 2} \tan^{-1}\left({\sin y\over \sinh x}\right).
\end{displaymath} (85)

Trigonometric half-angle formulas include

$\displaystyle \sin\left({\alpha\over 2}\right)$ $\textstyle =$ $\displaystyle \sqrt{1-\cos\alpha\over 2}$ (86)
$\displaystyle \cos\left({\alpha\over 2}\right)$ $\textstyle =$ $\displaystyle \sqrt{1+\cos\alpha\over 2}$ (87)
$\displaystyle \tan\left({\alpha\over 2}\right)$ $\textstyle =$ $\displaystyle {\sin\alpha\over 1+\cos\alpha}$ (88)
  $\textstyle =$ $\displaystyle {1-\cos\alpha\over\sin\alpha}$ (89)
  $\textstyle =$ $\displaystyle {1\pm\sqrt{1+\tan^2\alpha}\over\tan\alpha}$ (90)
  $\textstyle =$ $\displaystyle {\tan\alpha\sin\alpha\over\tan\alpha+\sin\alpha}.$ (91)

The Prosthaphaeresis Formulas are

$\displaystyle \sin\alpha+\sin\beta$ $\textstyle =$ $\displaystyle 2\sin[{\textstyle{1\over 2}}(\alpha+\beta)]\cos[{\textstyle{1\over 2}}(\alpha-\beta)]$ (92)
$\displaystyle \sin\alpha-\sin\beta$ $\textstyle =$ $\displaystyle 2\cos[{\textstyle{1\over 2}}(\alpha+\beta)]\sin[{\textstyle{1\over 2}}(\alpha-\beta)]$ (93)
$\displaystyle \cos\alpha+\cos\beta$ $\textstyle =$ $\displaystyle 2\cos[{\textstyle{1\over 2}}(\alpha+\beta)]\cos[{\textstyle{1\over 2}}(\alpha-\beta)]$ (94)
$\displaystyle \cos\alpha-\cos\beta$ $\textstyle =$ $\displaystyle -2\sin[{\textstyle{1\over 2}}(\alpha+\beta)]\sin[{\textstyle{1\over 2}}(\alpha-\beta)].$ (95)

Related formulas are
$\displaystyle \sin\alpha\cos\beta$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\sin(\alpha-\beta)+\sin(\alpha+\beta)]$ (96)
$\displaystyle \cos\alpha\cos\beta$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]$ (97)
$\displaystyle \cos\alpha\sin\beta$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\sin(\alpha+\beta)-\sin(\alpha-\beta)]$ (98)
$\displaystyle \sin\alpha\sin\beta$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\cos(\alpha-\beta)-\cos(\alpha+\beta)].$ (99)

Multiplying both sides by 2 gives the equations sometimes known as the Werner Formulas.

Trigonometric product/sum formulas are

\sin(\alpha+\beta)\sin(\alpha-\beta) = \sin^2\alpha-\sin^2\beta = \cos^2\beta-\cos^2\alpha
\end{displaymath} (100)

\cos(\alpha+\beta)\cos(\alpha-\beta) = \cos^2\alpha-\sin^2\beta = \cos^2\beta-\sin^2\alpha.
\end{displaymath} (101)

Power formulas include

$\displaystyle \sin^2 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[1-\cos(2x)]$ (102)
$\displaystyle \sin^3 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[3\sin x-\sin(3x)]$ (103)
$\displaystyle \sin^4 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}[3-4\cos(2x)+\cos(4x)]$ (104)

$\displaystyle \cos^2 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[1+\cos(2x)]$ (105)
$\displaystyle \cos^3 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[3\cos x+\cos(3x)]$ (106)
$\displaystyle \cos^4 x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}[3+4\cos(2x)+\cos(4x)]$ (107)

(Beyer 1987, p. 140). Formulas of these types can also be given analytically as

$\displaystyle \sin^{2n}x$ $\textstyle =$ $\displaystyle {1\over 2^{2n}}{2n\choose n}+{(-1)^n\over 2^{2n-1}}\sum_{k=0}^{n-1} (-1)^k{2n\choose k}\cos[2(n-k)x]\quad\eqnum$ (108)
$\displaystyle \sin^{2n+1}$ $\textstyle =$ $\displaystyle {(-1)^n\over 4^n}\sum_{k=0}^n (-1)^k{2n+1\choose k}\sin[(2n+1-2k)x]$ (109)
$\displaystyle \cos^{2n}x$ $\textstyle =$ $\displaystyle {1\over 2^{2n}}{2n\choose n}+{1\over 2^{2n-1}}\sum_{k=0}^{n-1}{2n\choose k}\cos[2(n-k)x]$ (110)
$\displaystyle \cos^{2n+1}x$ $\textstyle =$ $\displaystyle {1\over 4^n}\sum_{k=0}^n{2n+1\choose k}\cos[(2n+1-2k)x]$ (111)

(Kogan), where ${n\choose m}$ is a Binomial Coefficient.

Trigonometric identities which prove useful in the construction of map projections include

$=\sin(2\phi)(A'+\cos(2\phi)(B'+\cos(2\phi)(C'+D'\cos(2\phi)))),\quad$ (112)

$\displaystyle A'$ $\textstyle \equiv$ $\displaystyle A-C$ (113)
$\displaystyle B'$ $\textstyle \equiv$ $\displaystyle 2B-4D$ (114)
$\displaystyle C'$ $\textstyle \equiv$ $\displaystyle 4C$ (115)
$\displaystyle D'$ $\textstyle \equiv$ $\displaystyle 8D.$ (116)

$ =\sin\phi(A'+\sin^2\phi(B'+\sin^2\phi(C'+D'\sin^2\phi))),\quad$ (117)
$\displaystyle A'$ $\textstyle \equiv$ $\displaystyle A+3B+5C+7D$ (118)
$\displaystyle B'$ $\textstyle \equiv$ $\displaystyle -4B-20C-56D$ (119)
$\displaystyle C'$ $\textstyle \equiv$ $\displaystyle 16C+112D$ (120)
$\displaystyle D'$ $\textstyle \equiv$ $\displaystyle -64D.$ (121)

$ =A'+\cos(2\phi)(B'+\cos(2\phi)(C'+\cos(2\phi)(D'+E'\cos(2\phi)))),\quad$ (122)


$\displaystyle A'$ $\textstyle \equiv$ $\displaystyle A-C+E$ (123)
$\displaystyle B'$ $\textstyle \equiv$ $\displaystyle B-3D$ (124)
$\displaystyle C'$ $\textstyle \equiv$ $\displaystyle 2C-8E$ (125)
$\displaystyle D'$ $\textstyle \equiv$ $\displaystyle 4D$ (126)
$\displaystyle E'$ $\textstyle \equiv$ $\displaystyle 8E$ (127)

(Snyder 1987).

See also Cosecant, Cosine, Cotangent, Euclidean Number, Inverse Cosecant, Inverse Cosine, Inverse Cotangent, Inverse Secant, Inverse Sine, Inverse Tangent, Inverse Trigonometric Functions, Osborne's Rule, Polygon, Secant, Sine, Tangent, Trigonometry Values: Pi, Trigonometry Values: Pi/2, Trigonometry Values: Pi/3, Trigonometry Values: Pi/4, Trigonometry Values: Pi/5, Trigonometry Values: Pi/6, Trigonometry Values: Pi/7, Trigonometry Values: Pi/8, Trigonometry Values: Pi/9, Trigonometry Values: Pi/10, Trigonometry Values: Pi/11, Trigonometry Values: Pi/12, Trigonometry Values: Pi/15, Trigonometry Values: Pi/16, Trigonometry Values: Pi/17, Trigonometry Values: Pi/18, Trigonometry Values: Pi/20, Trigonometry Values 0, Werner Formulas



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Bahm, L. B. The New Trigonometry on Your Own. Patterson, NJ: Littlefield, Adams & Co., 1964.

Beyer, W. H. ``Trigonometry.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 134-152, 1987.

Dixon, R. ``The Story of Sine and Cosine.'' § 4.4 in Mathographics. New York: Dover, pp. 102-106, 1991.

Hobson, E. W. A Treatise on Plane Trigonometry. London: Cambridge University Press, 1925.

Kells, L. M.; Kern, W. F.; and Bland, J. R. Plane and Spherical Trigonometry. New York: McGraw-Hill, 1940.

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Kung, S. H. ``Proof Without Words: The Difference-Product Identities'' and ``Proof Without Words: The Sum-Product Identities.'' Math. Mag. 69, 269, 1996.

Maor, E. Trigonometric Delights. Princeton, NJ: Princeton University Press, 1998.

Morrill, W. K. Plane Trigonometry, rev. ed. Dubuque, IA: Wm. C. Brown, 1964.

Robinson, R. M. ``A Curious Mathematical Identity.'' Amer. Math. Monthly 64, 83-85, 1957.

Sloane, N. J. A. Sequence A003401/M0505 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 19, 1987.

Thompson, J. E. Trigonometry for the Practical Man. Princeton, NJ: Van Nostrand.

mathematica.gif Weisstein, E. W. ``Exact Values of Trigonometric Functions.'' Mathematica notebook TrigExact.m.

Yates, R. C. ``Trigonometric Functions.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 225-232, 1952.

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