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Trigonometric function

From Simple English Wikipedia, the free encyclopedia
All of the trigonometric functions of anyangle can beconstructed using acircle centered atO withradius of 1.
Trigonometric functions:Sine,Cosine,Tangent,Cosecant,Secant,Cotangent

Inmathematics, thetrigonometric functions are a set of functions which relateangles to the sides of aright triangle. There are many trigonometric functions, the 3 most common being sine, cosine,tangent, followed bycotangent,secant andcosecant.[1][2] The last three are calledreciprocal trigonometric functions, because they act as thereciprocals of other functions. Secant and cosecant are rarely used.

FunctionAbbreviationRelation (Radians)
Sinesinsin⁡θ=cos⁡(π2−θ){\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)\,}
Cosinecoscos⁡θ=sin⁡(π2−θ){\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)\,}
Tangenttan
(or tg)		tan⁡θ=sin⁡θcos⁡θ=cot⁡(π2−θ)=1cot⁡θ{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}\,}
Cotangentcot
(or ctg)		cot⁡θ=cos⁡θsin⁡θ=tan⁡(π2−θ)=1tan⁡θ{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}\,}
Secantsecsec⁡θ=1cos⁡θ=csc⁡(π2−θ){\displaystyle \sec \theta ={\frac {1}{\cos \theta }}=\csc \left({\frac {\pi }{2}}-\theta \right)\,}
Cosecantcsc
(or cosec)		csc⁡θ=1sin⁡θ=sec⁡(π2−θ){\displaystyle \csc \theta ={\frac {1}{\sin \theta }}=\sec \left({\frac {\pi }{2}}-\theta \right)\,}

Definition

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Thetrigonometric functions sometimes are also called circular functions. They arefunctions of anangle; they are important when studyingtriangles, among many other applications. Trigonometric functions are commonly defined asratios of two sides of a right triangle containing the angle,[3] and can equivalently be defined as the lengths of various line segments from aunit circle (a circle with radius of one).

Right triangle definitions

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Aright triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle.

In order to define the trigonometric functions for the angleA, start with aright triangle that contains the angleA:

We use the following names for the sides of the triangle:

  • Thehypotenuse is the side opposite the right angle, also the longest side of a right-angled triangle, in this caseh.
  • Theopposite side is the side opposite to the angle we are interested in, in this casea.
  • Theadjacent side is the side that is in contact with the right angle the angle we are interested in, hence its name. In this case, the adjacent side isb.

All triangles are taken to exist inEuclidean geometry, so that the inside angles of each triangle sum to πradians (or 180°); therefore, for a right triangle, the two non-right angles are between zero and π/2 radians. Notice that strictly speaking, the following definitions only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by using theunit circle, or by requiring certain symmetries and that they beperiodic functions.

1) Thesine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case[3]

sin⁡A=oppositehypotenuse=ah.{\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}.}

Note that since all those triangles aresimilar, this ratio does not depend on the particular right triangle that is chosen, as long as it contains the angleA.

The set of zeroes of sine (that is, the values ofx{\displaystyle x} for whichsin⁡x=0{\displaystyle \sin x=0}) is

{nπ|n∈Z}.{\displaystyle \left\{n\pi {\big |}n\in \mathbb {Z} \right\}.}

2) Thecosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case[3]

cos⁡A=adjacenthypotenuse=bh.{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}.}

The set of zeroes of cosine is

{π2+nπ|n∈Z}.{\displaystyle \left\{{\frac {\pi }{2}}+n\pi {\bigg |}n\in \mathbb {Z} \right\}.}

3) Thetangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case[3]

tan⁡A=oppositeadjacent=ab.{\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}.}

The set of zeroes of tangent is

{nπ|n∈Z}.{\displaystyle \left\{n\pi {\big |}n\in \mathbb {Z} \right\}.}

This is the same set as that of the sine function, since

tan⁡A=sin⁡Acos⁡A.{\displaystyle \tan A={\frac {\sin A}{\cos A}}.}

The remaining three functions are best defined using the above three functions.

4) Thecosecant csc(A) is themultiplicative inverse of sin(A); it is the ratio of the length of the hypotenuse to the length of the opposite side:[3]

csc⁡A=hypotenuseopposite=ha{\displaystyle \csc A={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}={\frac {h}{a}}}.

5) Thesecant sec(A) is themultiplicative inverse of cos(A); it is the ratio of the length of the hypotenuse to the length of the adjacent side:[3]

sec⁡A=hypotenuseadjacent=hb{\displaystyle \sec A={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}={\frac {h}{b}}}.

6) Thecotangent cot(A) is themultiplicative inverse of tan(A); it is the ratio of the length of the adjacent side to the length of the opposite side:

cot⁡A=adjacentopposite=ba{\displaystyle \cot A={\frac {\textrm {adjacent}}{\textrm {opposite}}}={\frac {b}{a}}}.

Definitions by power series

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One can also define the trigonometric functions by usingpower series:

sin⁡x=x−x33!+x55!−x77!+⋯=∑n=0∞(−1)nx2n+1(2n+1)!{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}
cos⁡x=1−x22!+x44!−x66!+⋯=∑n=0∞(−1)nx2n(2n)!{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}[4]

and define tangent, cotangent, secant and cosecant using identities, see below.

Identities

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Some importantidentities:

tan⁡x=sin⁡xcos⁡x{\displaystyle \tan x={\frac {\sin x}{\cos x}}}
cot⁡x=cos⁡xsin⁡x{\displaystyle \cot x={\frac {\cos x}{\sin x}}}
sec⁡x=1cos⁡x{\displaystyle \sec x={\frac {1}{\cos x}}}
csc⁡x=1sin⁡x{\displaystyle \csc x={\frac {1}{\sin x}}}
sin2⁡x+cos2⁡x=1{\displaystyle \sin ^{2}x+\cos ^{2}x=1}
sin⁡2x=2sin⁡xcos⁡x{\displaystyle \sin 2x=2\sin x\cos x}
cos⁡2x=cos⁡xcos⁡x−sin⁡xsin⁡x=cos2⁡x−sin2⁡x=2cos2⁡x−1=1−2sin2⁡x{\displaystyle \cos 2x=\cos x\cos x-\sin x\sin x=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x}
tan⁡2x=2tan⁡x1−tan2⁡x{\displaystyle \tan 2x={\frac {2\tan x}{1-\tan ^{2}x}}}
sin⁡(x±y)=sin⁡xcos⁡y±cos⁡xsin⁡y{\displaystyle \sin \left(x\pm y\right)=\sin x\cos y\pm \cos x\sin y}
cos⁡(x±y)=cos⁡xcos⁡y∓sin⁡xsin⁡y{\displaystyle \cos \left(x\pm y\right)=\cos x\cos y\mp \sin x\sin y}
tan⁡(x±y)=tan⁡x±tan⁡y1∓tan⁡xtan⁡y{\displaystyle \tan \left(x\pm y\right)={\frac {\tan x\pm \tan y}{1\mp \tan x\tan y}}}

Hyperbolic functions

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Thehyperbolic functions are like the trigonometric functions, in that they have very similar properties. Each of six trigonometric functions has a corresponding hyperbolic form.[1] They are defined in terms of theexponential function, which is based on the constante.

  • Hyperbolic sine:
sinh⁡x=ex−e−x2=e2x−12ex=1−e−2x2e−x.{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}
  • Hyperbolic cosine:
cosh⁡x=ex+e−x2=e2x+12ex=1+e−2x2e−x.{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}
  • Hyperbolic tangent:
tanh⁡x=sinh⁡xcosh⁡x=ex−e−xex+e−x=e2x−1e2x+1=1−e−2x1+e−2x.{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}.}
  • Hyperbolic cotangent:
coth⁡x=cosh⁡xsinh⁡x=ex+e−xex−e−x=e2x+1e2x−1=1+e−2x1−e−2x,x≠0.{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}},\qquad x\neq 0.}
  • Hyperbolic secant:
sechx=1cosh⁡x=2ex+e−x=2exe2x+1=2e−x1+e−2x.{\displaystyle \operatorname {sech} \,x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}.}
  • Hyperbolic cosecant:
cschx=1sinh⁡x=2ex−e−x=2exe2x−1=2e−x1−e−2x,x≠0.{\displaystyle \operatorname {csch} \,x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}},\qquad x\neq 0.}

Related pages

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References

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  1. ↑1.01.1"Comprehensive List of Algebra Symbols".Math Vault. 2020-03-25. Retrieved2020-08-29.
  2. ↑Weisstein, Eric W."Trigonometric Functions".mathworld.wolfram.com. Retrieved2020-08-29.
  3. ↑3.03.13.23.33.43.5"Sine, Cosine, Tangent".www.mathsisfun.com. Retrieved2020-08-29.
  4. ↑Weisstein, Eric W."Cosine".mathworld.wolfram.com. Retrieved2020-08-29.

Bibliography

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