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| The '''Pythagorean trigonometric identity''' is a [[trigonometric identity]] expressing the [[Pythagorean theorem]] in terms of [[trigonometric function]]s. Along with the [[trigonometric identity#Angle sum and difference identities|sum-of-angles formulae]], it is one of the basic relations between the [[sine]] and [[cosine]] functions, from which [[trigonometric functions|all others]] may be derived.
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| ==Statement of the identity==
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| {{See also|Trigonometric functions|List of trigonometric identities}}
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| Mathematically, the Pythagorean identity states:
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| :<math>\sin^2 \theta + \cos^2 \theta = 1.\!</math>
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| (Note that {{nowrap|sin<sup>2</sup> ''θ''}} means {{nowrap|(sin ''θ'')<sup>2</sup>)}}. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.<ref name= Leff>
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| {{Cite book|title=PreCalculus the Easy Way |author=Lawrence S. Leff |url=http://books.google.com/books?id=y_7yrqrHTb4C&pg=PA296 |page=296 |isbn=0-7641-2892-2 |edition=7th |publisher=Barron's Educational Series |year=2005}}
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| </ref>
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| If the length of the hypotenuse of a [[right triangle]] is 1, then the lengths of the legs are the [[sine]] and [[cosine]] of one of the angles. Therefore, this trigonometric identity follows from the [[Pythagorean theorem]].
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| ==Proofs and their relationships to the Pythagorean theorem==
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| [[File:Trig Functions.PNG|thumb |Similar right triangles showing sine and cosine of angle θ]]
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| ===Proof based on right-angle triangles===
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| Any [[similar triangles]] have the property that if we select the same angle in all them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.
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| The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:
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| :<math>\sin \theta = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}= \frac{b}{c}</math>
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| :<math>\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}} = \frac{a}{c} \ .</math>
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| The Pythagorean identity follows by squaring both definitions above, and adding; the [[left-hand side]] of the identity then becomes
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| :<math>\frac{\mathrm{opposite}^2 + \mathrm{adjacent}^2}{\mathrm{hypotenuse}^2}</math>
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| which by the Pythagorean theorem is equal to 1. Note, however, that this definition is valid only for angles between 0 and π/2 radians (not inclusive) and therefore this argument does not prove the identity for all angles. Values of 0 and π/2 are trivially proven by direct evaluation of sin and cos at those angles.
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| To complete the proof, the identities found at [[trigonometric identity#Symmetry, shifts, and periodicity|Trigonometric symmetry, shifts, and periodicity]] may be employed. By the periodicity identities we can say if the formula is true for {{nowrap|−π < ''θ'' ≤ π}} then it is true for all real ''θ''. Next we prove the range {{nowrap|π/2 < ''θ'' ≤ π,}} to do this we let {{nowrap|''t'' {{=}} ''θ'' − π/2,}} ''t'' will now be in the range {{nowrap|0 < ''t'' ≤ π/2.}} We can then make use of squared versions of some basic shift identities (squaring conveniently removes the minus signs):
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| : <math>\sin^2\theta+\cos^2\theta = \sin^2\left(t+\frac{1}{2}\pi\right) + \cos^2\left(t+\frac{1}{2}\pi\right) = \cos^2t+\sin^2t = 1.</math>
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| All that remains is to prove it for {{nowrap|−π < ''θ'' < 0;}} this can be done by squaring the symmetry identities to get
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| : <math>\sin^2\theta=\sin^2(-\theta)\text{ and }\cos^2\theta=\cos^2(-\theta).\,</math>
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| ====Related identities====
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| [[File:Trig Functions 2.PNG|thumb|Similar right triangles illustrating the tangent and secant trigonometric functions.]]
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| The identities
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| :<math>1 + \tan^2 \theta = \sec^2 \theta\,</math>
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| and
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| :<math>1 + \cot^2 \theta = \csc^2 \theta\,</math>
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| are also called Pythagorean trigonometric identities.<ref name= Leff/> If one leg of a right triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg, and the secant of the angle is the length of the hypotenuse.
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| :<math> \tan \theta =\frac{b}{a} \ , </math>
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| and:
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| :<math> \sec \theta = \frac{c}{a} \ . </math>
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| In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.
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| =====Tabulation of derivations=====
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| Another way of thinking about the other identities is to derive them from the original identity. The following table shows how this is done by dividing each element of the original Pythagorean Identity by a common divisor.
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| {| class="wikitable sortable"
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| !Original Identity
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| !Divisor
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| !Divisor Equation
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| !Derived Identity
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| !Derived Identity (Alternate)
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| |-
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| | <math>\sin^2 \theta + \cos^2 \theta = 1\!</math> || || || ||
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| |-
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| | <math>\sin^2 \theta + \cos^2 \theta = 1\!</math> || <MATH> \cos^2 \theta \!</MATH> || <math> \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}\!</math>
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| || <math> \tan^2 \theta + 1 = \sec^2 \theta\!</math> || <math> \tan^2 \theta = \sec^2 \theta - 1\!</math>
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| |-
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| | <math>\sin^2 \theta + \cos^2 \theta = 1\!</math> || <MATH> \sin^2 \theta \!</MATH> || <math> \frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}\!</math>
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| || <math> 1 + \cot^2 \theta = \csc^2 \theta\!</math> || <math> \cot^2 \theta = \csc^2 \theta - 1\!</math>
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| |-
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| |}
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| ===Proof using the unit circle===
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| {{Main|unit circle}}
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| [[File:Trig functions on unit circle.PNG|thumb|Point ''P''(x,y) on the circle of unit radius at an [[obtuse angle]] θ > π/2]]
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| [[File:Periodic sine.PNG|thumb|Sine function on unit circle (top) and its graph (bottom)]]
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| The unit circle centered at the origin in the Euclidean plane is defined by the equation:<ref name=circle>
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| This result can be found using the distance formula <math>d = \sqrt{x^2 +y^2}\,</math> for the distance from the origin to the point <math>(x,\ y)\,</math>. See {{Cite book|title=Algebra and Trigonometry |author=Cynthia Y. Young |url=http://books.google.com/books?id=ARmvHS83xf0C&pg=PA210 |page=210 |isbn=0-470-22273-5 |year=2009 |edition=2nd |publisher=Wiley}} This approach assumes Pythagoras' theorem. Alternatively, one could simply substitute values and determine that the graph is a circle.
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| </ref>
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| :<math>x^2 + y^2 = 1.\,</math> | |
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| Given an angle θ, there is a unique point ''P'' on the unit circle at an angle θ from the ''x''-axis, and the ''x''- and ''y''-coordinates of ''P'' are:<ref name=Shaw>
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| {{Cite book|title=Contemporary Precalculus: A Graphing Approach |author=[[Thomas W. Hungerford]], Douglas J. Shaw |url=http://books.google.com/books?id=esexVzMJwIMC&pg=PA442 |chapter=§6.2 The sine, cosine and tangent functions |page=442 |edition=5th |year=2008 |publisher=Cengage Learning |isbn=0-495-10833-2}}
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| </ref>
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| :<math>x = \cos \theta \ \mathrm { and} \ y = \sin \theta \ .</math>
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| Consequently, from the equation for the unit circle:
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| :<math> \cos^2 \theta + \sin^2 \theta = 1 \ , </math>
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| the Pythagorean identity.
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| In the figure, the point ''P'' has a ''negative'' x-coordinate, and is appropriately given by ''x'' = cos''θ'', which is a negative number: cos''θ'' = −cos(π−''θ'' ). Point ''P'' has a positive ''y''-coordinate, and sin''θ'' = sin(π−''θ'' ) > 0. As ''θ'' increases from zero to the full circle ''θ'' = 2π, the sine and cosine change signs in the various quadrants to keep ''x'' and ''y'' with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant.
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| Because the ''x''- and ''y''-axes are perpendicular, this Pythagorean identity is actually equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1 (which is in turn equivalent to the full Pythagorean theorem by applying a similar-triangles argument). See [[unit circle]] for a short explanation.
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| ===Proof using power series===
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| The trigonometric functions may also be defined using [[power series]], namely (for ''x'' an angle measured in [[radian]]s):<ref name=Hamilton>
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| {{Cite book|title=Time series analysis |author=James Douglas Hamilton |url=http://books.google.com/books?id=B8_1UBmqVUoC&pg=PA714 |page=714 |chapter=Power series
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| |isbn=0-691-04289-6 |year=1994 |publisher=Princeton University Press}}
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| </ref><ref name=Krantz>
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| {{Cite book|title=Real analysis and foundations |author=Steven George Krantz |url=http://books.google.com/books?id=oWao9tYvYXAC&pg=PA269 |pages=269–270
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| |isbn=1-58488-483-5 |chapter=Definition 10.3 |year=2005 |publisher=CRC Press |edition=2nd}}
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| </ref>
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| :<math>\begin{align}
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| \sin x &= \sum_{n = 0}^\infty \frac{(-1)^n}{(2n + 1)!} x^{2n + 1},\\
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| \cos x &= \sum_{n = 0}^\infty \frac{(-1)^n}{(2n)!} x^{2n}.
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| \end{align}</math>
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| Using the formal multiplication law for power series at [[power series#Multiplication and division|Multiplication and division of power series]] (suitably modified to account for the form of the series here) we obtain
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| : <math>
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| \begin{align}
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| \sin^2 x & = \sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{(-1)^i}{(2i + 1)!} \frac{(-1)^j}{(2j + 1)!} x^{(2i + 1) + (2j + 1)} \\
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| & = \sum_{n = 1}^\infty \left(\sum_{i = 0}^{n - 1} \frac{(-1)^{n - 1}}{(2i + 1)!(2(n - i - 1) + 1)!}\right) x^{2n} \\
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| & = \sum_{n = 1}^\infty \left( \sum_{i = 0}^{n - 1} {2n \choose 2i + 1} \right) \frac{(-1)^{n - 1}}{(2n)!} x^{2n},\\
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| \cos^2 x & = \sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{(-1)^i}{(2i)!} \frac{(-1)^j}{(2j)!} x^{(2i) + (2j)} \\
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| & = \sum_{n = 0}^\infty \left(\sum_{i = 0}^n \frac{(-1)^n}{(2i)!(2(n - i))!}\right) x^{2n} \\
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| & = \sum_{n = 0}^\infty \left( \sum_{i = 0}^n {2n \choose 2i} \right) \frac{(-1)^n}{(2n)!} x^{2n}.
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| \end{align}
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| </math>
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| Note that in the expression for sin<sup>2</sup>, ''n'' must be at least 1, while in the expression for cos<sup>2</sup>, the [[constant term]] is equal to 1. The remaining terms of their sum are (with common factors removed)
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| :<math>\sum_{i = 0}^n {2n \choose 2i} - \sum_{i = 0}^{n - 1} {2n \choose 2i + 1}
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| = \sum_{j = 0}^{2n} (-1)^j {2n \choose j}
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| = (1 - 1)^{2n}
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| = 0 </math>
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| by the [[binomial theorem]]. Consequently,
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| :<math>\sin^2 x + \cos^2 x = 1 \ , </math>
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| which is the Pythagorean trigonometric identity.
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| The Pythagorean theorem is not closely related to the Pythagorean identity when the trigonometric functions are defined in this way; instead, in combination with the theorem, the identity now shows that these power series [[parametric plot|parameterize]] the unit circle, which we used in the previous section. Note that this definition actually constructs the sin and cos functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.
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| ===Proof using the differential equation===
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| Sine and Cosine [[trigonometric function#Definitions via differential equations|can be defined]] as the two solutions to the differential equation:<ref name=Debnath>
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| {{Cite book|title=Linear partial differential equations for scientists and engineers |author=Tyn Myint U., Lokenath Debnath |url=http://books.google.com/books?id=Zbz5_UvERIIC&pg=PA316 |page=316 |chapter=Example 8.12.1 |isbn=0-8176-4393-1 |edition=4th |publisher=Springer |year=2007}}
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| </ref>
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| ::<math>y'' + y = 0\,</math>
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| satisfying respectively ''y''(0) = 0, ''y''′(0) = 1 and ''y''(0) = 1, ''y''′(0) = 0. It follows from the theory of [[ordinary differential equation]]s that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that the function
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| :<math>z = \sin^2 x + \cos^2 x\,</math>
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| is constant and equal to 1. Differentiating using the [[chain rule]] gives:
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| :<math> \frac{d}{dx} z = 2 \sin x \ \cos x + 2 \cos x \ (-\sin x) = 0 \ , </math>
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| so ''z'' is constant. A calculation confirms that ''z''(0) = 1, and ''z'' is a constant so ''z'' = 1 for all ''x'', so the Pythagorean identity is established.
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| A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine. In fact, the definitions by ordinary differential equation and by power series lead to similar derivations of most identities.
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| This proof of the identity has no direct connection with Euclid's demonstration of the Pythagorean theorem.
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| ==See also==
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| *[[Pythagorean theorem]]
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| *[[Trigonometric identity]]
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| *[[Unit circle]]
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| *[[Power series]]
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| *[[Differential equation]]
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| ==In-line notes and references==
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| <references/>
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| ==External links==
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| * [http://sympl.org/book/examples/interactive-plots/pythagorean-identity/ An interactive illustration of Pythagorean identity]
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| {{DEFAULTSORT:Pythagorean Trigonometric Identity}}
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| [[Category:Mathematical identities]]
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| [[Category:Articles containing proofs]]
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| [[Category:Trigonometry]]
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| [[Category:Pythagoras|Identity]]
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| [[ar:متطابقة فيثاغورث]]
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| [[da:Idiotformlen]]
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| [[no:Enhetsformelen]]
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| [[pt:Identidade trigonométrica fundamental]]
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