त्रिकोणमितीय सर्वसमिकाओं की सूची

मुक्त ज्ञानकोश विकिपीडिया से
यहाँ जाएँ: भ्रमण, खोज

त्रिकोणमितीय सर्वसमिकाएँ, चरों के त्रिकोणमितीय फलनों के रूप में व्यक्त समतायें होती हैं।

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

अनुक्रम

प्रयुक्त संकेतों (Notation) के बारे में[संपादित करें]

To avoid the confusion caused by the ambiguity of sin−1(x), the reciprocals and inverses of trigonometric functions are often displayed as in this table. In representing the cosecant function, the longer form 'cosec' is sometimes used in place of 'csc'.

Function Inverse function Reciprocal Inverse reciprocal
sine sin arcsine arcsin cosecant csc arccosecant arccsc
cosine cos arccosine arccos secant sec arcsecant arcsec
tangent tan arctangent arctan cotangent cot arccotangent arccot

Different angular measures can be appropriate in different situations. This table shows some of the more common systems. Radians is the default angular measure and is the one you use if you use the exponential definitions. All angular measures are unitless.

Degrees 30 45 60 90 120 180 270 360
Radians \pi/6 \pi/4 \pi/3 \pi/2 2\pi/3 \pi 3\pi/2 2\pi
Grads 33 ⅓ 50 66 ⅔ 100 133 ⅓ 200 300 400

मौलिक सम्बन्ध[संपादित करें]

Pythagorean trigonometric identity \sin^2 \theta + \cos^2 \theta = 1\,[1]
Ratio identity \tan \theta = \frac{\sin \theta}{\cos \theta}[1]

From the two identities above, the following table can be extrapolated. Note however that these conversion equations may not provide the correct sign (+ or −). For example, if sin θ = 1/2, the conversion in the table indicates that \scriptstyle\cos\theta\,=\,\sqrt{1 - \sin^2\theta} = \sqrt{3}/2, though it is possible that \scriptstyle\cos\theta \,=\, -\sqrt{3}/2. More information would be needed about which quadrant θ lies in to determine a single, exact answer.

Each trigonometric function in terms of the other five.
Function sin cos tan cosec sec cot
\sin \theta =  \sin \theta\  \sqrt{1 - \cos^2\theta}  \frac{\tan\theta}{\sqrt{1 + \tan^2\theta}} पार्स नहीं कर पाये (अज्ञात फंक्शन '\cosec'): \frac{1}{\cosec \theta}  \frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}  \frac{1}{\sqrt{1+\cot^2\theta}}
\cos \theta =  \sqrt{1 - \sin^2\theta}  \cos \theta\  \frac{1}{\sqrt{1 + \tan^2 \theta}} पार्स नहीं कर पाये (अज्ञात फंक्शन '\cosec'): \frac{\sqrt{\cosec^2\theta - 1}}{\cosec \theta}  \frac{1}{\sec \theta}  \frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}
\tan \theta =  \frac{\sin\theta}{\sqrt{1 - \sin^2\theta}}  \frac{\sqrt{1 - \cos^2\theta}}{\cos \theta}  \tan \theta\  \frac{1}{\sqrt{\csc^2\theta - 1}}  \sqrt{\sec^2\theta - 1}  \frac{1}{\cot \theta}
\csc \theta =  {1 \over \sin \theta}  {1 \over \sqrt{1 - \cos^2 \theta}}  {\sqrt{1 + \tan^2\theta} \over \tan \theta}  \csc \theta\  {\sec \theta \over \sqrt{\sec^2\theta - 1}}  \sqrt{1 + \cot^2 \theta}
\sec \theta =  {1 \over \sqrt{1 - \sin^2\theta}}  {1 \over \cos \theta}  \sqrt{1 + \tan^2\theta}  {\csc\theta \over \sqrt{\csc^2\theta - 1}} \sec\theta\  {\sqrt{1 + \cot^2\theta} \over \cot \theta}
\cot \theta =  {\sqrt{1 - \sin^2\theta} \over \sin \theta}  {\cos \theta \over \sqrt{1 - \cos^2\theta}}  {1 \over \tan\theta}  \sqrt{\csc^2\theta - 1}  {1 \over \sqrt{\sec^2\theta - 1}}  \cot\theta\

ऐतिहासिक पृष्टभूमि[संपादित करें]

Rarely used today, the versine, coversine, haversine, and exsecant have been defined as below and used in navigation, for example the haversine formula was used to calculate the distance between two points on a sphere.

नाम मान
\textrm{versin} \, \theta  1 - \cos \theta \,
\textrm{coversin} \, \theta 1 - \sin \theta \,
\textrm{haversin} \, \theta \tfrac{1}{2} \textrm{versin} \theta \,
\textrm{exsec} \, \theta \,  \sec \theta - 1 \,

सममिति, अनतर एवं आवर्तन (Symmetry, shifts, and periodicity)[संपादित करें]

इकाई वृत्त (unit circle) का परीक्षण करके हम त्रिकोणमित्तीय फलनों के निम्नलिखित गुणों को स्थापित कर सकते हैं:

सममिति (Symmetry)[संपादित करें]

When the trigonometric functions are reflected from certain values of \theta, The result is often one of the other trigonometric functions. This leads to the following identities:

Reflected in \theta=0 Reflected in \theta= \pi/2 Reflected in \theta= \pi

\begin{align}
\sin(0-\theta) &= -\sin \theta \\
\cos(0-\theta) &= +\cos \theta \\
\tan(0-\theta) &= -\tan \theta \\
\csc(0-\theta) &= -\csc \theta \\
\sec(0-\theta) &= +\sec \theta \\
\cot(0-\theta) &= -\cot \theta
\end{align}

\begin{align}
\sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\
\cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\
\tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\
\csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\
\sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\
\cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta
\end{align}

\begin{align}
\sin(\pi - \theta) &= +\sin \theta \\
\cos(\pi - \theta) &= -\cos \theta \\
\tan(\pi - \theta) &= -\tan \theta \\
\csc(\pi - \theta) &= +\csc \theta \\
\sec(\pi - \theta) &= -\sec \theta \\
\cot(\pi - \theta) &= -\cot \theta \\
\end{align}

आवर्तता (periodicity)[संपादित करें]

By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express the result more simply. Some examples of this are given shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.

Shift by π/2 Shift by π
Period for tan and cot
Shift by 2π
Period for sin, cos, csc and sec

\begin{align}
\sin(\theta + \tfrac{\pi}{2}) &= +\cos \theta \\
\cos(\theta + \tfrac{\pi}{2}) &= -\sin \theta \\
\tan(\theta + \tfrac{\pi}{2}) &= -\cot \theta \\
\csc(\theta + \tfrac{\pi}{2}) &= +\sec \theta \\
\sec(\theta + \tfrac{\pi}{2}) &= -\csc \theta \\
\cot(\theta + \tfrac{\pi}{2}) &= -\tan \theta
\end{align}

\begin{align}
\sin(\theta + \pi) &= -\sin \theta \\
\cos(\theta + \pi) &= -\cos \theta \\
\tan(\theta + \pi) &= +\tan \theta \\
\csc(\theta + \pi) &= -\csc \theta \\
\sec(\theta + \pi) &= -\sec \theta \\
\cot(\theta + \pi) &= +\cot \theta \\
\end{align}

\begin{align}
\sin(\theta + 2\pi) &= +\sin \theta \\
\cos(\theta + 2\pi) &= +\cos \theta \\
\tan(\theta + 2\pi) &= +\tan \theta \\
\csc(\theta + 2\pi) &= +\csc \theta \\
\sec(\theta + 2\pi) &= +\sec \theta \\
\cot(\theta + 2\pi) &= +\cot \theta
\end{align}

कोणों के योग और अन्तर की सर्वसमिकाएँ (Angle sum and difference identities)[संपादित करें]

इनको योग एवं अन्तर के सूत्र भी कहते हैं। इनको यूलर के सूत्र की सहायता से बडी आसानी से सिद्ध किया जा सकता है।

Sine \sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi \,[2] Note: From Plus-minus sign#Minus-plus sign.

\begin{align}x \pm y = a \pm b &\Rightarrow \ x + y = a + b \\ &\mbox{and} \ x -y = a -b \end{align}
\begin{align}
x \pm y = a \mp b &\Rightarrow \ x + y = a - b \\ &\mbox{and}\  x - y = a + b\end{align}

Cosine \cos(\theta \pm \phi) = \cos \theta \cos \phi \mp \sin \theta \sin \phi\,[2]
Tangent \tan(\theta \pm \phi) = \frac{\tan \theta \pm \tan \phi}{1 \mp \tan \theta \tan \phi}[2]

अनन्त पदों के योग की ज्या एवं कोज्या[संपादित करें]

(Sines and cosines of sums of infinitely many terms)

 \sin\left(\sum_{i=1}^\infty \theta_i\right)
=\sum_{\mathrm{odd}\  k \ge 1} (-1)^{(k-1)/2}
\sum_{ |A| = k }
\left(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\right)
 \cos\left(\sum_{i=1}^\infty \theta_i\right)
=\sum_{\mathrm{even}\  k \ge 0} ~ (-1)^{k/2} ~~
\sum_{ |A| = k }
\left(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\right)

where "|A| = k" means the index A runs through the set of all subsets of size k of the set { 1, 2, 3, ... }.

In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors.

If only finitely many of the terms θi are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.

अनेक किन्तु सीमित पदों के योग की स्पर्शज्या[संपादित करें]

(Tangents of sums of finitely many terms)

Let xi = tan(θi ), for i = 1, ..., n. Let ek be the kth-degree elementary symmetric polynomial in the variables xi, i = 1, ..., n, k = 0, ..., n. Then

\tan(\theta_1+\cdots+\theta_n) = \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots},

the number of terms depending on n.

उदाहरण्के लिये,

 \begin{align} \tan(\theta_1 + \theta_2 + \theta_3)
&{}= \frac{e_1 - e_3}{e_0 - e_2} = \frac{(x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3)}{
1 \ - \ (x_1 x_2 + x_1 x_3 + x_2 x_3)}, \\  \\
\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4)
&{}= \frac{e_1 - e_3}{e_0 - e_2 + e_4} \\  \\
&{}= \frac{(x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4)}{
1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4)},\end{align}

आदि आदि। इसकी सामान्य स्थिति (general case) गणितीय निगमन (mathematical induction) की सहायता से सिद्ध किया जा सकता है।

कोणों के गुणज के त्रिकोणमित्तीय अनुपात[संपादित करें]

(Multiple-angle formulae)

\cos n\theta =T_n (\cos \theta)\,[3] Tn nवाँ चेविशेव बहुपद (Chebyshev polynomial) है।
\sin^2 n\theta = S_n (\sin^2\theta)\, Sn nवाँ spread polynomial है।
\cos n\theta +i\sin n\theta=(\cos(\theta)+i\sin(\theta))^n \, द मायवर का सूत्र (de Moivre's formula), i काल्पनिक इकाई (Imaginary unit) है।
1+2\cos(x) + 2\cos(2x) + 2\cos(3x) + \cdots + 2\cos(nx)
= \frac{\sin\left(\left(n +\frac{1}{2}\right)x\right)}{\sin(x/2)}.

(x के इस फलन (फंक्शन) को डिरिचलेट कर्नेल (Dirichlet kernel) कहते हैं।

दोगुना, तिगुना एवं आधे कोण के सूत्र[संपादित करें]

(Double-, triple-, and half-angle formulae)

इन्हें योग एवं अन्तर की सर्वसमिकाओं की सहायता से या गुणज-कोण की सर्वसमिकाओं की सहायता से प्रदर्शित (सिद्ध) किया जा सकता है।

Double-angle formulae[4]
\begin{align}
\sin 2\theta &= 2 \sin \theta \cos \theta \ \\ &= \frac{2 \tan \theta} {1 + \tan^2 \theta}
\end{align} \begin{align}
\cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ &= 2 \cos^2 \theta - 1 \\ 
&= 1 - 2 \sin^2 \theta \\ &= \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}
\end{align} \tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\, \cot 2\theta = \frac{\cot \theta - \tan \theta}{2}\,
Triple-angle formulae[3]
\sin 3\theta = 3 \sin \theta- 4 \sin^3\theta \, \cos 3\theta = 4 \cos^3\theta - 3 \cos \theta \, \tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}  
Half-angle formulae[5]
\sin \tfrac{\theta}{2} =  \pm\, \sqrt{\frac{1 - \cos \theta}{2}} \cos \tfrac{\theta}{2} =  \pm\, \sqrt{\frac{1 + \cos\theta}{2}} \begin{align} \tan \tfrac{\theta}{2} &= \csc \theta - \cot \theta \\ &= \pm\, \sqrt{1 - \cos \theta \over 1 + \cos \theta} \\ &= \frac{\sin \theta}{1 + \cos \theta} \\ &= \frac{1-\cos \theta}{\sin \theta} \end{align} \cot \tfrac{\theta}{2} = \csc \theta + \cot \theta

See also Tangent half-angle formula.

यूलर का अनन्त गुणनफल[संपादित करें]

(Euler's infinite product )

 \cos\left({\theta \over 2}\right) \cdot \cos\left({\theta \over 4}\right)
\cdot \cos\left({\theta \over 8}\right)\cdots = \prod_{n=1}^\infty \cos\left({\theta \over 2^n}\right)
= {\sin(\theta)\over \theta}.

घातांक कम करने का सूत्र[संपादित करें]

(Power-reduction formulae)

इन्हें कोज्या द्वि-गुण कोण सूत्र के द्वीतीय एवं तृतीय रूप का प्रयोग करके प्राप्त किया जा सकता है।

Sine \sin^2\theta = \frac{1 - \cos 2\theta}{2} \sin^3\theta = \frac{3 \sin\theta - \sin 3\theta}{4}
Cosine \cos^2\theta = \frac{1 + \cos 2\theta}{2} \cos^3\theta = \frac{3 \cos\theta + \cos 3\theta}{4}
Other \sin^2\theta \cos^2\theta = \frac{1 - \cos 4\theta}{8} \sin^3\theta \cos^3\theta = \frac{\sin^3 2\theta}{8}

गुनफल-से-योग तथा योग-से-गुणनफल की सर्वसमिकाएँ[संपादित करें]

(Product-to-sum and sum-to-product identities)

Product-to-sum
\cos \theta \cos \phi = {\cos(\theta - \phi) + \cos(\theta + \phi) \over 2}
\sin \theta \sin \phi = {\cos(\theta - \phi) - \cos(\theta + \phi) \over 2}
\sin \theta \cos \phi = {\sin(\theta + \phi) + \sin(\theta - \phi) \over 2}
Sum-to-product
\sin \theta + \sin \phi = 2 \sin\left(\frac{\theta + \phi}{2} \right) \cos\left(\frac{\theta - \phi}{2} \right)
\cos \theta + \cos \phi = 2 \cos\left(\frac{\theta + \phi} {2} \right) \cos\left(\frac{\theta - \phi}{2} \right)
\cos \theta - \cos \phi = -2\sin\left({\theta + \phi \over 2}\right) \sin\left({\theta - \phi \over 2}\right)
\sin \theta - \sin \phi = 2 \cos\left({\theta + \phi \over 2}\right) \sin\left({\theta - \phi\over 2}\right) \;

अन्य सम्बन्धित सर्वसमिकाएँ[संपादित करें]

If x, y, and z are the three angles of any triangle, or in other words

\mbox{if }x + y + z = \pi = \mbox{half circle,}\,
\mbox{then }\tan(x) + \tan(y) + \tan(z) = \tan(x)\tan(y)\tan(z).\,

(If any of x, y, z is a right angle, one should take both sides to be ∞. This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by tan(θ) as tan(θ) either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)

\mbox{If }x + y + z = \pi = \mbox{half circle,}\,
\mbox{then }\sin(2x) + \sin(2y) + \sin(2z) = 4\sin(x)\sin(y)\sin(z).\,

टोलेमी का प्रमेय (Ptolemy's theorem)[संपादित करें]

 \mbox{If }w + x + y + z = \pi = \mbox{half circle,} \,
\begin{align} \mbox{then }
&     \sin(w + x)\sin(x + y) \\
&{} = \sin(x + y)\sin(y + z) \\
&{} = \sin(y + z)\sin(z + w) \\
&{} = \sin(z + w)\sin(w + x) = \sin(w)\sin(y) + \sin(x)\sin(z).
\end{align}

(The first three equalities are trivial; the fourth is the substance of this identity.) Essentially this is Ptolemy's theorem adapted to the language of trigonometry.

Linear combinations[संपादित करें]

For some purposes it is important to know that any linear combination of sine waves of the same period but different phase shifts is also a sine wave with the same period, but a different phase shift. In the case of a linear combination of a sine and cosine wave, we have

a\sin x+b\cos x=\sqrt{a^2+b^2}\cdot\sin(x+\varphi)\,

where

\varphi = \arctan \left(\frac{b}{a}\right)

More generally, for an arbitrary phase shift, we have

a\sin x+b\sin(x+\alpha)= c \sin(x+\beta)\,

where


  c = \sqrt{a^2 + b^2 +2ab\cos \alpha},

and


  \beta = {\rm arctan} \left(\frac{b\sin \alpha}{a + b\cos \alpha}\right).
  • note: arcsin, arccos, arctan are all inverses.

Other sums of trigonometric functions[संपादित करें]

Sum of sines and cosines with arguments in arithmetic progression:

\sin{\varphi} + \sin{(\varphi + \alpha)} + \sin{(\varphi + 2\alpha)} + 
\cdots + \sin{(\varphi + n\alpha)}=\frac{\sin{\left(\frac{(n+1) \alpha}{2}\right)} \cdot \sin{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}.
\cos{\varphi} + \cos{(\varphi + \alpha)} + \cos{(\varphi + 2\alpha)} + 
\cdots + \cos{(\varphi + n\alpha)}=\frac{\sin{\left(\frac{(n+1) \alpha}{2}\right)} \cdot \cos{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}.

For any a and b:

a \cos(x) + b \sin(x) = \sqrt{ a^2 + b^2 } \cos(x - \arctan(b, a)) \;

where arctan(y, x) is the generalization of arctan(y/x) which covers the entire circular range (see also the account of this same identity in "symmetry, periodicity, and shifts" above for this generalization of arctan).

\tan(x) + \sec(x) = \tan\left({x \over 2} + {\pi \over 4}\right).

The above identity is sometimes convenient to know when thinking about the Gudermanian function.

If x, y, and z are the three angles of any triangle, i.e. if x + y + z = π, then

\cot(x)\cot(y) + \cot(y)\cot(z) + \cot(z)\cot(x) = 1.\,

व्युत्क्रम त्रिकोणमित्तीय फलन[संपादित करें]

(Inverse trigonometric functions)

 \arcsin(x)+\arccos(x)=\pi/2\;
 \arctan(x)+\arccot(x)=\pi/2.\;
\arctan(x)+\arctan(1/x)=\left\{\begin{matrix} \pi/2, & \mbox{if }x > 0 \\  -\pi/2, & \mbox{if }x < 0 \end{matrix}\right.

Compositions of trig and inverse trig functions[संपादित करें]

\sin[\arccos(x)]=\sqrt{1-x^2} \,
\sin[\arctan(x)]=\frac{x}{\sqrt{1+x^2}}
\cos[\arctan(x)]=\frac{1}{\sqrt{1+x^2}}
\cos[\arcsin(x)]=\sqrt{1-x^2} \,
\tan[\arcsin (x)]=\frac{x}{\sqrt{1 - x^2}}
\tan[\arccos (x)]=\frac{\sqrt{1 - x^2}}{x}

Relation to the complex exponential function[संपादित करें]

e^{ix} = \cos(x) + i\sin(x)\, (Euler's formula),
e^{-ix} = \cos(-x) + i\sin(-x) = \cos(x) - i\sin(x)\,
e^{i\pi} = -1\,
\cos(x) = \frac{e^{ix} + e^{-ix}}{2} \;
\sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \;

where i² = −1.

"cis"[संपादित करें]

साँचा:Split-section Occasionally one sees the notation

\operatorname{cis}(x) = \cos(x) + i\sin(x),\,

i.e. "cis" abbreviates "cos + i sin".

Though at first glance this notation is redundant, being equivalent to eix, its use is rooted in several advantages.

Convenience[संपादित करें]

This notation was more common in the post WWII era when typewriters were used to convey mathematical expressions. Superscripts are both offset vertically and smaller than 'cis' or 'exp'; hence, they can be problematic even for hand writing. For example e ix² versus cis(x²) versus exp(ix²). For many readers, cis(x²) is the clearest, easiest to read of the three.

The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis and cos + i sin notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin).

The cis notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding doesn't yet permit the notation e ix. As students learn concepts that build on prior knowledge, it is important not to force them into levels of math they are not yet prepared for.

Pedagogy[संपादित करें]

In some contexts, the cis notation may serve the pedagogical purpose of emphasizing that one has not yet proved that this is an exponential function. In doing trigonometry without complex numbers, one may prove the two identities

\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y) = c_1 c_2 - s_1 s_2,\,
\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) = s_1 c_2 + c_1 s_2.\,

Similarly in treating multiplication of complex numbers (with no involvement of trigonometry), one may observe that the real and imaginary parts of the product of c1 + is1 and c2 + is2 are respectively

c_1 c_2 - s_1 s_2,\,
s_1 c_2 + c_1 s_2.\,

Thus one sees this same pattern arising in two disparate contexts:

  • trigonometry without complex numbers, and
  • complex numbers without trigonometry.

This coincidence can serve as a motivation for conjoining the two contexts and thereby discovering the trigonometric identity

\operatorname{cis}(x+y) = \operatorname{cis}(x)\operatorname{cis}(y),\,

and observing that this identity for cis of a sum is simpler than the identities for sin and cos of a sum. Having proved this identity, one can challenge the students to recall which familiar sort of function satisfies this same functional equation

f(x+y) = f(x)f(y).\,

The answer is exponential functions. That suggests that cis may be an exponential function

\operatorname{cis}(x) = b^x.\,

Then the question is: what is the base b? The definition of cis and the local behavior of sin and cos near zero suggest that

\operatorname{cis}(0+dx) = \operatorname{cis}(0) + i\,dx,

(where dx is an infinitesimal increment of x). Thus the rate of change at 0 is i, so the base should be ei. Thus if this is an exponential function, then it must be

\operatorname{cis}(x) = e^{ix}.\,

अनन्त गुणनफल का सूत्र[संपादित करें]

(Infinite product formula)

For applications to special functions, the following infinite product formulæ for trigonometric functions are useful:

\sin x = x \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2 n^2}\right)
\sinh x = x \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2 n^2}\right)
\frac{\sin x}{x} = \prod_{n = 1}^\infty\cos\left(\frac{x}{2^n}\right)
\cos x = \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2(n - \frac{1}{2})^2}\right)
\cosh x = \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2(n - \frac{1}{2})^2}\right)

The Gudermannian function[संपादित करें]

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers; see that article for details.

Identities without variables[संपादित करें]

The curious identity

\cos 20^\circ\cdot\cos 40^\circ\cdot\cos 80^\circ=\frac{1}{8}

is a special case of an identity that contains one variable:

\prod_{j=0}^{k-1}\cos(2^j x)=\frac{\sin(2^k x)}{2^k\sin(x)}.

A similar-looking identity is

 \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7} = \frac{1}{8},

and in addition

\sin 20^\circ\cdot\sin 40^\circ\cdot\sin 80^\circ=\sqrt{3}/8.

The following is perhaps not as readily generalized to an identity containing variables:

\cos 24^\circ+\cos 48^\circ+\cos 96^\circ+\cos 168^\circ=\frac{1}{2}.

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

 \cos\left(   \frac{2\pi}{21}\right)
  \,+\, \cos\left(2\cdot\frac{2\pi}{21}\right)  
  \,+\, \cos\left(4\cdot\frac{2\pi}{21}\right)
 
  \,+\, \cos\left(5\cdot\frac{2\pi}{21}\right)
  \,+\, \cos\left(8\cdot\frac{2\pi}{21}\right)
  \,+\, \cos\left(10\cdot\frac{2\pi}{21}\right)=\frac{1}{2}.

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

An efficient way to compute π is based on the following identity without variables, due to Machin:

\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}

or, alternatively, by using Euler's formula:

\frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79}.



\begin{matrix}
\sin 0 & = & \sin 0^\circ & = & 0 & = & \cos 90^\circ &  =  & \cos \left(\frac {\pi} {2} \right) \\  \\
\sin \left(\frac {\pi} {6} \right) & = & \sin 30^\circ & = & 1/2 & = & \cos 60^\circ & = & \cos \left(\frac {\pi} {3} \right) \\  \\
\sin \left(\frac {\pi} {4} \right) & = & \sin 45^\circ & = & \sqrt{2}/2 & = & \cos 45^\circ & = & \cos \left(\frac {\pi} {4} \right) \\  \\
\sin \left(\frac {\pi} {3} \right) & = & \sin 60^\circ & = & \sqrt{3}/2 & = & \cos 30^\circ & = & \cos \left(\frac {\pi} {6} \right) \\  \\
\sin \left(\frac {\pi} {2} \right) & = & \sin 90^\circ & = & 1 & = & \cos 0^\circ & = & \cos 0
\end{matrix}
\sin{\frac{\pi}{7}}=\frac{\sqrt{7}}{6}-
\frac{\sqrt{7}}{189} \sum_{j=0}^{\infty} \frac{(3j+1)!}{189^j j!\,  (2j+2)!}
\!
\sin{\frac{\pi}{18}}=
\frac{1}{6} \sum_{j=0}^{\infty} \frac{(3j)!}{27^j j!\,  (2j+1)!}
\!

With the golden ratio φ:

\cos \left(\frac {\pi} {5} \right) = \cos 36^\circ={\sqrt{5}+1 \over 4} = \varphi /2
\sin \left(\frac {\pi} {10} \right) = \sin 18^\circ = {\sqrt{5}-1 \over 4}  = {\varphi - 1 \over 2} = {1 \over 2\varphi}

Also see exact trigonometric constants.

चलन-कलन (Calculus)[संपादित करें]

In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, their derivatives can be found by verifying two limits. The first is:

\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1,

verified using the unit circle and squeeze theorem. It may be tempting to propose to use L'Hôpital's rule to establish this limit. However, if one uses this limit in order to prove that the derivative of the sine is the cosine, and then uses the fact that the derivative of the sine is the cosine in applying L'Hôpital's rule, one is reasoning circularly—a logical fallacy. The second limit is:

\lim_{x\rightarrow 0}\frac{1-\cos(x)}{x}=0,

verified using the identity tan(x/2) = (1 − cos(x))/sin(x). Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that sin′(x) = cos(x) and cos′(x) = −sin(x). If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

{d \over dx}\sin(x) = \cos(x)

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation:


\begin{matrix}
{d \over dx} \sin x =& \cos x         , & {d \over dx} \arcsin x =&  {1 \over \sqrt{1 - x^2}     } \\  \\
{d \over dx} \cos x =& -\sin x        , & {d \over dx} \arccos x =& {-1 \over \sqrt{1 - x^2}}      \\  \\
{d \over dx} \tan x =& \sec^2 x       , & {d \over dx} \arctan x =& { 1 \over 1 + x^2}            \\  \\
{d \over dx} \cot x =& -\csc^2 x      , & {d \over dx} \arccot x =& {-1 \over 1 + x^2}             \\  \\
{d \over dx} \sec x =& \tan x \sec x  , & {d \over dx} \arcsec x =& { 1 \over |x|\sqrt{x^2 - 1}}   \\  \\
{d \over dx} \csc x =& -\csc x \cot x , & {d \over dx} \arccsc x =& {-1 \over |x|\sqrt{x^2 - 1}}
\end{matrix}
[6]

The integral identities can be found in "list of integrals of trigonometric functions".

Implications[संपादित करें]

The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and fourier transformations.

चरघातांकी वाली परिभाषाएँ (Exponential definitions)[संपादित करें]

फलन (Function) व्युत्क्रम फलन (Inverse Function)
\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \, \arcsin x = -i \ln \left(ix + \sqrt{1 - x^2}\right) \,
\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \, \arccos x = -i \ln \left(x + \sqrt{x^2 - 1}\right) \,
\tan \theta = \frac{e^{i\theta} - e^{-i\theta}}{i(e^{i\theta} + e^{-i\theta})} \, \arctan x = \frac{i \ln \left(\frac{i + x}{i - x}\right)}{2} \,
\csc \theta = \frac{2i}{e^{i\theta} - e^{-i\theta}} \, \arccsc x = -i \ln \left(\tfrac{i}{x} + \sqrt{1 - \tfrac{1}{x^2}}\right) \,
\sec \theta = \frac{2}{e^{i\theta} + e^{-i\theta}} \, \arcsec x = -i \ln \left(\tfrac{1}{x} + \sqrt{1 - \tfrac{i}{x^2}}\right) \,
\cot \theta = \frac{i(e^{i\theta} + e^{-i\theta})}{e^{i\theta} - e^{-i\theta}} \, \arccot x = \frac{i \ln \left(\frac{i - x}{i + x}\right)}{2} \,
\operatorname{cis} \, \theta = e^{i\theta} \, \operatorname{arccis} \, x = \frac{\ln x}{i} \,

विविध[संपादित करें]

डिरिच्लेट् कर्नेल् (Dirichlet kernel)[संपादित करें]

The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:

1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots+2\cos(nx) = \frac{ \sin\left(\left(n+\frac{1}{2}\right)x\right) }{ \sin(x/2) }.

The convolution of any integrable function of period 2π with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any measure or generalized function.

अर्ध कोणों के सूत्रों का परिवर्धन (Extension of half-angle formulae)[संपादित करें]

यदि हम

t = \tan\left(\frac{x}{2}\right),

रखें तो,

    \sin(x) = \frac{2t}{1 + t^2}   and   \cos(x) = \frac{1 - t^2}{1 + t^2}   and   e^{i x} = \frac{1 + i t}{1 - i t}.

जहाँ eix और cis(x) एक ही चीज हैं।

उपरोक्त सूत्र कैलकुलस में sin(x) और cos(x) के rational functions के समाकलन करने में उपयोगी होते हैं।

इन्हें भी देखें[संपादित करें]

सन्दर्भ[संपादित करें]

बाहरी कड़ियाँ[संपादित करें]