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In mathematics , trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities , which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function , and then simplifying the resulting integral with a trigonometric identity. Several different units of angle measure are widely used, including degree , radian , and gradian gons :. The following table shows for some common angles their conversions and the values of the basic trigonometric functions:.

Results for other angles can be found at Trigonometric constants expressed in real radicals. The functions sine , cosine and tangent of an angle are sometimes referred to as the primary or basic trigonometric functions. The parentheses around the argument of the functions are often omitted, e. The sine of an angle is defined, in the context of a right triangle , as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle the hypotenuse.

The cosine of an angle in this context is the ratio of the length of the side that is adjacent to the angle divided by the length of the hypotenuse. The tangent of an angle in this context is the ratio of the length of the side that is opposite to the angle divided by the length of the side that is adjacent to the angle.

This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of sin and cos from above:. The remaining trigonometric functions secant sec , cosecant csc , and cotangent cot are defined as the reciprocal functions of cosine, sine, and tangent, respectively.

Rarely, these are called the secondary trigonometric functions:. These definitions are sometimes referred to as ratio identities. The inverse trigonometric functions are partial inverse functions for the trigonometric functions.

In trigonometry, the basic relationship between the sine and the cosine is given by the Pythagorean identity:. This equation can be solved for either the sine or the cosine:. Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other up to a plus or minus sign :. The versine , coversine , haversine , and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere.

They are rarely used today. By examining the unit circle, the following properties of the trigonometric functions can be established. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x -axis. These are also known as reduction formulae. By shifting the arguments of trigonometric functions by certain angles, it is sometimes possible that changing the sign or applying complementary trigonometric functions can express particular results more simply.

Some examples of shifts are shown below in the table. These are also known as the angle addition and subtraction theorems or formulae. The identities can be derived by combining right triangles such as in the adjacent diagram, or by considering the invariance of the length of a chord on a unit circle given a particular central angle. The most intuitive derivation uses rotation matrices see below.

Consequently, as the opposing sides of the diagram's outer rectangle are equal, we deduce. Relocating one of the named angles yields a variant of the diagram that demonstrates the angle difference formulae for sine and cosine.

These identities have applications in, for example, in-phase and quadrature components. In terms of rotation matrices :. The matrix inverse for a rotation is the rotation with the negative of the angle. These formulae show that these matrices form a representation of the rotation group in the plane technically, the special orthogonal group SO 2 , since the composition law is fulfilled and inverses exist. Since multiplication by a complex number of unit length rotates the complex plane by the argument of the number, the above multiplication of rotation matrices is equivalent to a multiplication of complex numbers:.

In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

Furthermore, in each term all but finitely many of the cosine factors are unity. The case of only finitely many terms can be proved by mathematical induction. Formulae for twice an angle. Formulae for triple angles. These can be shown by using either the sum and difference identities or the multiple-angle formulae.

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation , which allows one to prove that trisection is in general impossible using the given tools, by field theory. However, the discriminant of this equation is positive, so this equation has three real roots of which only one is the solution for the cosine of the one-third angle.

None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots. In each of these two equations, the first parenthesized term is a binomial coefficient , and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed.

The ratio of these formulae gives. It follows by induction that cos nx is a polynomial of cos x , the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials Trigonometric definition. The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems.

See amplitude modulation for an application of the product-to-sum formulae, and beat acoustics and phase detector for applications of the sum-to-product formulae.

Charles Hermite demonstrated the following identity. The first three equalities are trivial rearrangements; the fourth is the substance of this identity. For coprime integers n , m. More generally [36]. For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift.

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude, [37] [38]. The general case reads [38]. See also Phasor addition. These identities, named after Joseph Louis Lagrange , are: [39] [40]. A related function is the following function of x , called the Dirichlet kernel. The above identity is sometimes convenient to know when thinking about the Gudermannian function , which relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.

If x , y , and z are the three angles of any triangle, i. If f x is given by the linear fractional transformation. These formulae are useful for proving many other trigonometric identities. That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine.

The equality of the imaginary parts gives an angle addition formula for sine. For applications to special functions , the following infinite product formulae for trigonometric functions are useful: [46] [47]. In terms of the arctangent function we have [42]. The curious identity known as Morrie's law ,. The following is perhaps not as readily generalized to an identity containing variables but see explanation below :.

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively. Other cosine identities include: [48]. Many of those curious identities stem from more general facts like the following: [49]. The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles.

By setting the frequency as the cutoff frequency, the following identity can be proved:. Generally, for numbers t 1 , This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t 1 , In particular, the computed t n will be rational whenever all the t 1 , With these values,.

Also see trigonometric constants expressed in real radicals. Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:. Ptolemy used this proposition to compute some angles in his table of chords.

This identity involves a trigonometric function of a trigonometric function: [51]. In calculus the relations stated below require angles to be measured in radians ; the relations would become more complicated if angles were measured in another unit such as degrees.

If the trigonometric functions are defined in terms of geometry, along with the definitions of arc length and area , their derivatives can be found by verifying two limits. The first is:. The second limit is:. If the sine and cosine functions are defined by their Taylor series , then the derivatives can be found by differentiating the power series term-by-term. The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation : [52] [53] [54].

The integral identities can be found in List of integrals of trigonometric functions. Some generic forms are listed below. The fact that the differentiation of trigonometric functions sine and cosine results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms. This identity was discovered as a by-product of research in medical imaging.


List of trigonometric identities



Formule trigonometrie



Calcul avec les nombres complexes/Écriture exponentielle et trigonométrique


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