The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an
identity
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expressing the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
in terms of
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and cosine functions.
The identity is
:
As usual, means .
Proofs and their relationships to the Pythagorean theorem
Proof based on right-angle triangles
Any
similar triangles
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...
have the property that if we select the same angle in all of 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.
The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:
:
:
The Pythagorean identity follows by squaring both definitions above, and adding; the
left-hand side
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.Trigonometric symmetry, shifts, and periodicity may be employed. By the periodicity identities we can say if the formula is true for then it is true for all real ''θ''. Next we prove the range to do this we let ''t'' will now be in the range We can then make use of squared versions of some basic shift identities (squaring conveniently removes the minus signs):
:
All that remains is to prove it for this can be done by squaring the symmetry identities to get
:
Related identities
The identities
:
and
:
are also called Pythagorean trigonometric identities. 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.
:
and:
:
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.
The following table gives the identities with the factor or divisor that relates them to the main identity.
Proof using the unit circle
The unit circle centered at the origin in the Euclidean plane is defined by the equation:
This result can be found using the distance formula for the distance from the origin to the point . See This approach assumes Pythagoras' theorem. Alternatively, one could simply substitute values and determine that the graph is a circle.
:
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:
:
Consequently, from the equation for the unit circle:
:
the Pythagorean identity.
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.
Because the ''x''- and ''y''-axes are perpendicular, this Pythagorean identity is 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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
for a short explanation.
Proof using power series
The trigonometric functions may also be defined using
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
, namely (for ''x'' an angle measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s):
:
Using the formal multiplication law for power series at Multiplication and division of power series (suitably modified to account for the form of the series here) we obtain
:
In the expression for sin2, ''n'' must be at least 1, while in the expression for cos2, the
constant term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial
:x^2 + 2x + 3,\
the 3 is a constant term.
After like terms are com ...
is equal to 1. The remaining terms of their sum are (with common factors removed)
:
by the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Consequently,
:
which is the Pythagorean trigonometric identity.
When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.
Proof using the differential equation
Sine and cosine can be defined as the two solutions to the differential equation:
::
satisfying respectively and . It follows from the theory of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
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
:
is constant and equal to 1. Differentiating using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
gives:
:
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.
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.
This proof of the identity has no direct connection with Euclid's demonstration of the Pythagorean theorem.
Proof using Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
states that
:
So,
:.
See also
*
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
*
List of trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
*
Unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
*
Power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
*
Differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
Identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...