There are several equivalent ways for defining
trigonometric functions, and the proof of the
trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a
right angle. For greater and negative
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s, see
Trigonometric functions.
Other definitions, and therefore other proofs are based on the
Taylor series of
sine and
cosine, or on the
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 ...
to which they are solutions.
Elementary trigonometric identities
Definitions
The six trigonometric functions are defined for every
real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
:
:
:
:
:
:
Ratio identities
In the case of angles smaller than a right angle, the following identities are direct consequences of above definitions through the division identity
:
They remain valid for angles greater than 90° and for negative angles.
:
:
:
:
:
Or
:
:
Complementary angle identities
Two angles whose sum is π/2 radians (90 degrees) are ''complementary''. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:
:
:
:
:
:
:
Pythagorean identities
Identity 1:
:
The following two results follow from this and the ratio identities. To obtain the first, divide both sides of
by
; for the second, divide by
.
:
:
Similarly
:
:
Identity 2:
The following accounts for all three reciprocal functions.
:
Proof 2:
Refer to the triangle diagram above. Note that
by
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 ...
.
:
Substituting with appropriate functions -
:
Rearranging gives:
:
Angle sum identities
Sine
Draw a horizontal line (the ''x''-axis); mark an origin O. Draw a line from O at an angle
above the horizontal line and a second line at an angle
above that; the angle between the second line and the ''x''-axis is
.
Place P on the line defined by
at a unit distance from the origin.
Let PQ be a line perpendicular to line OQ defined by angle
, drawn from point Q on this line to point P.
OQP is a right angle.
Let QA be a perpendicular from point A on the ''x''-axis to Q and PB be a perpendicular from point B on the ''x''-axis to P.
OAQ and OBP are right angles.
Draw R on PB so that QR is parallel to the ''x''-axis.
Now angle
(because
,
making
, and finally
)
:
:
:
:
:
, so
:
, so
:
By substituting
for
and using
Symmetry, we also get:
:
:
Cosine
Using the figure above,
:
:
:
:
, so
:
, so
:
By substituting
for
and using
Symmetry, we also get:
:
:
Also, using the complementary angle formulae,
:
Tangent and cotangent
From the sine and cosine formulae, we get
:
Dividing both numerator and denominator by
, we get
:
Subtracting
from
, using
,
:
Similarly from the sine and cosine formulae, we get
:
Then by dividing both numerator and denominator by
, we get
:
Or, using
,
:
Using
,
:
Double-angle identities
From the angle sum identities, we get
:
and
:
The Pythagorean identities give the two alternative forms for the latter of these:
:
:
The angle sum identities also give
:
:
It can also be proved using
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 a ...
:
Squaring both sides yields
:
But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields
:
It follows that
:
.
Expanding the square and simplifying on the left hand side of the equation gives
:
.
Because the imaginary and real parts have to be the same, we are left with the original identities
:
,
and also
:
.
Half-angle identities
The two identities giving the alternative forms for cos 2θ lead to the following equations:
:
:
The sign of the square root needs to be chosen properly—note that if 2 is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
For the tan function, the equation is:
:
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
:
Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
:
This also gives:
:
Similar manipulations for the cot function give:
:
Miscellaneous – the triple tangent identity
If
half circle (for example,
,
and
are the angles of a triangle),
:
Proof:
:
Miscellaneous – the triple cotangent identity
If
quarter circle,
:
.
Proof:
Replace each of
,
, and
with their complementary angles, so cotangents turn into tangents and vice versa.
Given
:
:
so the result follows from the triple tangent identity.
Sum to product identities
*
*
*
Proof of sine identities
First, start with the sum-angle identities:
:
:
By adding these together,
:
Similarly, by subtracting the two sum-angle identities,
:
Let
and
,
:
and
Substitute
and
:
:
Therefore,
:
Proof of cosine identities
Similarly for cosine, start with the sum-angle identities:
:
:
Again, by adding and subtracting
:
:
Substitute
and
as before,
:
:
Inequalities
The figure at the right shows a sector of a circle with radius 1. The sector is of the whole circle, so its area is . We assume here that .
:
:
:
The area of triangle is , or . The area of triangle is , or .
Since triangle lies completely inside the sector, which in turn lies completely inside triangle , we have
:
This geometric argument relies on definitions of
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
and
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
, which act as assumptions, so it is rather a condition imposed in construction of
trigonometric functions than
a provable property.
[
] For the sine function, we can handle other values. If , then . But (because of the Pythagorean identity), so . So we have
:
For negative values of we have, by the symmetry of the sine function
:
Hence
:
and
:
Identities involving calculus
Preliminaries
:
:
Sine and angle ratio identity
:
In other words, the function sine is
differentiable at 0, and its
derivative is 1.
Proof: From the previous inequalities, we have, for small angles
:
,
Therefore,
:
,
Consider the right-hand inequality. Since
:
:
Multiply through by
:
Combining with the left-hand inequality:
:
Taking
to the limit as
:
Therefore,
:
Cosine and angle ratio identity
:
Proof:
:
The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.
Cosine and square of angle ratio identity
:
Proof:
As in the preceding proof,
:
The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.
Proof of compositions of trig and inverse trig functions
All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function
:
Proof:
We start from
:
(I)
Then we divide this equation (I) by
:
(II)
:
Then use the substitution
:
:
:
Then we use the identity
:
(III)
And initial Pythagorean trigonometric identity proofed...
Similarly if we divide this equation (I) by
:
(II)
:
Then use the substitution
:
:
Then we use the identity
:
(III)
And initial Pythagorean trigonometric identity proofed...
: