The differentiation of trigonometric functions is the mathematical process of finding the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a
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 a ...
, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(''a'') = cos(''a''), meaning that the rate of change of sin(''x'') at a particular angle ''x = a'' is given by the cosine of that angle.
All derivatives of circular trigonometric functions can be found from those of sin(''x'') and cos(''x'') by means of the
quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
applied to functions such as tan(''x'') = sin(''x'')/cos(''x''). Knowing these derivatives, the derivatives of the
inverse trigonometric functions
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
are found using
implicit differentiation
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...
.
Proofs of derivatives of trigonometric functions
Limit of sin(θ)/θ as θ tends to 0
The diagram at right shows a circle with centre ''O'' and radius ''r ='' 1. Let two radii ''OA'' and ''OB'' make an arc of θ radians. Since we are considering the limit as ''θ'' tends to zero, we may assume ''θ'' is a small positive number, say 0 < θ < ½ π in the first quadrant.
In the diagram, let ''R''
1 be the triangle ''OAB'', ''R''
2 the
circular sector
A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the large ...
''OAB'', and ''R''
3 the triangle ''OAC''. The
area of triangle ''OAB'' is:
:
The
area of the circular sector ''OAB'' is
, while the area of the triangle ''OAC'' is given by
:
Since each region is contained in the next, one has:
:
Moreover, since in the first quadrant, we may divide through by ½ , giving:
:
In the last step we took the reciprocals of the three positive terms, reversing the inequities.
We conclude that for 0 < θ < ½ π, the quantity is ''always'' less than 1 and ''always'' greater than cos(θ). Thus, as ''θ'' gets closer to 0, is "
squeezed" between a ceiling at height 1 and a floor at height , which rises towards 1; hence sin(''θ'')/''θ'' must tend to 1 as ''θ'' tends to 0 from the positive side:
For the case where ''θ'' is a small negative number –½ π < θ < 0, we use the fact that sine is an
odd function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
:
:
Limit of (cos(θ)-1)/θ as θ tends to 0
The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of ''θ'' is unimportant.
:
Using
the fact that the limit of a product is the product of limits, and the limit result from the previous section, we find that:
:
Limit of tan(θ)/θ as θ tends to 0
Using the limit for the
sine function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of limits, we find:
:
Derivative of the sine function
We calculate the derivative of the
sine function from the
limit definition:
:
Using the
angle addition formula , we have:
:
Using the limits for the
sine and
cosine functions:
:
Derivative of the cosine function
From the definition of derivative
We again calculate the derivative of the
cosine function
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 opp ...
from the limit definition:
:
Using the angle addition formula , we have:
:
Using the limits for the
sine and
cosine functions:
:
From the chain rule
To compute the derivative of the cosine function from the chain rule, first observe the following three facts:
:
:
:
The first and the second are
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 ...
, and the third is proven above. Using these three facts, we can write the following,
:
We can differentiate this 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 , ...
. Letting
, we have:
:
.
Therefore, we have proven that
:
.
Derivative of the tangent function
From the definition of derivative
To calculate the derivative of the
tangent 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 al ...
tan ''θ'', we use
first principles
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
. By definition:
:
Using the well-known angle formula , we have:
:
Using the fact that the limit of a product is the product of the limits:
:
Using the limit for the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
function, and the fact that tan ''δ'' tends to 0 as δ tends to 0:
:
We see immediately that:
:
From the quotient rule
One can also compute the derivative of the tangent function using the
quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
.
:
The numerator can be simplified to 1 by the
Pythagorean identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations be ...
, giving us,
:
Therefore,
:
Proofs of derivatives of inverse trigonometric functions
The following derivatives are found by setting a
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
''y'' equal to the
inverse trigonometric function
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
that we wish to take the derivative of. Using
implicit differentiation
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...
and then solving for ''dy''/''dx'', the derivative of the inverse function is found in terms of ''y''. To convert ''dy''/''dx'' back into being in terms of ''x'', we can draw a reference triangle on the unit circle, letting ''θ'' be y. Using the
Pythagorean theorem and the definition of the regular trigonometric functions, we can finally express ''dy''/''dx'' in terms of ''x''.
Differentiating the inverse sine function
We let
:
Where
:
Then
:
Taking the derivative with respect to
on both sides and solving for dy/dx:
:
:
Substituting
in from above,
:
Substituting
in from above,
:
:
Differentiating the inverse cosine function
We let
:
Where
:
Then
:
Taking the derivative with respect to
on both sides and solving for dy/dx:
:
:
Substituting
in from above, we get
:
Substituting
in from above, we get
:
:
Alternatively, once the derivative of
is established, the derivative of
follows immediately by differentiating the identity
so that
.
Differentiating the inverse tangent function
We let
:
Where
:
Then
:
Taking the derivative with respect to
on both sides and solving for dy/dx:
:
Left side:
:
using the Pythagorean identity
Right side:
:
Therefore,
:
Substituting
in from above, we get
:
:
Differentiating the inverse cotangent function
We let
:
where