Editing Trigonometric functions (section)

==Unit-circle definitions==
[[Image:Circle-trig6.svg|right|thumb|upright=1.35|All of the trigonometric functions of the angle {{math|''θ''}} (theta) can be constructed geometrically in terms of a unit circle centered at ''O''.]]
[[File:Periodic sine.svg|thumb|Sine function on unit circle (top) and its graph (bottom)]]
[[File:Unit Circle Definitions of Six Trigonometric Functions.svg|thumb|upright=1.2|In this illustration, the six trigonometric functions of an arbitrary angle {{math|''θ''}} are represented as [[Cartesian coordinates]] of points related to the [[unit circle]]. The {{mvar|y}}-axis ordinates of {{math|A}}, {{math|B}} and {{math|D}} are {{math|sin ''θ''}}, {{math|tan ''θ''}} and {{math|csc ''θ''}}, respectively, while the {{mvar|x}}-axis abscissas of {{math|A}}, {{math|C}} and {{math|E}} are {{math|cos ''θ''}}, {{math|cot ''θ''}} and {{math|sec ''θ''}}, respectively.]]
[[File:trigonometric function quadrant sign.svg|thumb|Signs of trigonometric functions in each quadrant. [[mnemonics in trigonometry|Mnemonics]] like "'''all''' '''s'''tudents '''t'''ake '''c'''alculus" indicates when '''s'''ine, '''c'''osine, and '''t'''angent are positive from quadrants I to IV.<ref name=steuben>{{Cite book |last1=Stueben |first1=Michael |title=Twenty years before the blackboard: the lessons and humor of a mathematics teacher |last2=Sandford |first2=Diane |date=1998 |publisher=Mathematical Association of America |isbn=978-0-88385-525-6 |series=Spectrum series |location=Washington, DC|page=119|url=https://books.google.com/books?id=qnd0P-Ja-O8C&dq=%22All+Students+Take+Calculus%22&pg=PA119}}</ref>]]

The six trigonometric functions can be defined as [[Cartesian coordinate system|coordinate values]] of points on the [[Euclidean plane]] that are related to the [[unit circle]], which is the [[circle]] of radius one centered at the origin {{math|O}} of this coordinate system. While [[#Right-angled triangle definitions|right-angled triangle definitions]] allow for the definition of the trigonometric functions for angles between {{math|0}} and <math display="inline">\frac{\pi}{2}</math> [[radian]]s {{math|(90°),}} the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let <math>\mathcal L</math> be the [[Ray (geometry)|ray]] obtained by rotating by an angle {{mvar|θ}} the positive half of the {{math|''x''}}-axis ([[counterclockwise]] rotation for <math>\theta > 0,</math> and clockwise rotation for <math>\theta < 0</math>). This ray intersects the unit circle at the point <math>\mathrm{A} = (x_\mathrm{A},y_\mathrm{A}).</math> The ray <math>\mathcal L,</math> extended to a [[line (geometry)|line]] if necessary, intersects the line of equation <math>x=1</math> at point <math>\mathrm{B} = (1,y_\mathrm{B}),</math> and the line of equation <math>y=1</math> at point <math>\mathrm{C} = (x_\mathrm{C},1).</math> The [[tangent line]] to the unit circle at the point {{math|A}}, is [[perpendicular]] to <math>\mathcal L,</math> and intersects the {{math|''y''}}- and {{math|''x''}}-axes at points <math>\mathrm{D} = (0,y_\mathrm{D})</math> and <math>\mathrm{E} = (x_\mathrm{E},0).</math> The [[Cartesian coordinates|coordinates]] of these points give the values of all trigonometric functions for any arbitrary real value of {{mvar|θ}} in the following manner.

The trigonometric functions {{math|cos}} and {{math|sin}} are defined, respectively, as the ''x''- and ''y''-coordinate values of point {{math|A}}. That is,
:<math>\cos \theta = x_\mathrm{A} \quad</math> and <math>\quad \sin \theta = y_\mathrm{A}.</math><ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Trigonometric_functions|title=Trigonometric Functions|last=Bityutskov|first=V.I.|date=7 February 2011|website=Encyclopedia of Mathematics|language=en|archive-url=https://web.archive.org/web/20171229231821/https://www.encyclopediaofmath.org/index.php/Trigonometric_functions|archive-date=29 December 2017|url-status=live|access-date=29 December 2017}}</ref>

In the range <math>0 \le \theta \le \pi/2</math>, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius {{math|OA}} as [[hypotenuse]]. And since the equation <math>x^2+y^2=1</math> holds for all points <math>\mathrm{P} = (x,y)</math> on the unit circle, this definition of cosine and sine also satisfies the [[Pythagorean identity]].
:<math>\cos^2\theta+\sin^2\theta=1.</math>

The other trigonometric functions can be found along the unit circle as 
:<math>\tan \theta = y_\mathrm{B} \quad</math> and <math> \quad\cot \theta = x_\mathrm{C},</math>
:<math>\csc \theta\ = y_\mathrm{D} \quad</math> and <math> \quad\sec \theta = x_\mathrm{E}.</math>

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
: <math>\tan \theta =\frac{\sin \theta}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \csc\theta=\frac{1}{\sin\theta}.</math>

[[File:Trigonometric functions.svg|right|thumb|upright=1.35|link={{filepath:trigonometric_functions_derivation_animation.svg}}|Trigonometric functions:
{{color|#00A|Sine}},
{{color|#0A0|Cosine}},
{{color|#A00|Tangent}},
{{color|#00A|Cosecant (dotted)}},
{{color|#0A0|Secant (dotted)}},
{{color|#A00|Cotangent (dotted)}}
– [{{filepath:trigonometric_functions_derivation_animation.svg}} animation] ]]

Since a rotation of an angle of <math>\pm2\pi</math> does not change the position or size of a shape, the points {{math|A}}, {{math|B}}, {{math|C}}, {{math|D}}, and {{math|E}} are the same for two angles whose difference is an integer multiple of <math>2\pi</math>. Thus trigonometric functions are [[periodic function]]s with period <math>2\pi</math>. That is, the equalities
: <math> \sin\theta = \sin\left(\theta + 2 k \pi \right)\quad</math> and <math>\quad \cos\theta = \cos\left(\theta + 2 k \pi \right)</math>
hold for any angle {{mvar|θ}} and any [[integer]] {{mvar|k}}. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that <math>2\pi</math> is the smallest value for which they are periodic (i.e., <math>2\pi</math> is the [[periodic function|fundamental period]] of these functions). However, after a rotation by an angle <math>\pi</math>, the points {{mvar|B}} and {{mvar|C}} already return to their original position, so that the tangent function and the cotangent function have a fundamental period of <math>\pi</math>. That is, the equalities
: <math> \tan\theta = \tan(\theta + k\pi) \quad</math> and <math>\quad \cot\theta = \cot(\theta + k\pi)</math>
hold for any angle {{mvar|θ}} and any integer {{mvar|k}}.