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2.3.2 Angle Transformation Formulae
ОглавлениеReferring again to Fig. 2.10a, one may label as u1 the unit vector centered at the origin, defining an angle θ1 with the horizontal axis – with coordinates (cos θ1, sin θ1) as per Eqs. (2.288) and (2.290); and likewise as u2 the unit vector centered at O but defining an angle θ2 – with coordinates (cos θ2, sin θ2), with θ2 > θ1 for simplicity. Under these circumstances, the scalar product of u1 and u2 (to be discussed later) reads
as per its defining algorithm, and because ‖ u1‖ = ‖ u2‖ = 1 by hypothesis; Eq. (2.317) readily simplifies to
where θ2 − θ1 > 0 represents the amplitude of the angle defined by vectors u1 and u2, i.e. ∠ u1, u2. As will be duly proven below, u1 · u2 may instead be calculated via
so elimination of u1· u2 between Eqs. (2.318) and (2.319) unfolds
this is equivalent to
after having relabeled θ1 and θ2 to y and x, respectively. Equation (2.321), known as the basic angle transformation formula, permits calculation of the cosine of a difference of angles based on knowledge of sine and cosine of the individual angles – and is actually valid, irrespective of the relative amplitude of angles θ1 and θ2. If θ2 is set equal to π/2 in particular, then Eq. (2.320) becomes
(2.322)
which reduces to
because cos π/2 is nil and sin π/2 is unity; Eq. (2.323) confirms that sine and cosine are complementary functions – so a change of variable to x ≡ π/2 − θ1 (implying θ1 = π/2 − x) allows retrieval of Eq. (2.294), as expected.
After rewriting Eq. (2.321) as
(2.324)
at the expense of Eqs. (2.295) and (2.296), and changing notation of −y to y (in view of its being a dummy variable), one obtains
Eq. (2.325) permits rapid calculation of the cosine of a sum of two arguments, again based on the sine and cosine of the individual arguments. On the other hand, Eqs. (2.294) and (2.323) allow reformulation of Eq. (2.321) to
(2.326)
where the argument of the left‐hand side may be rearranged to read
since dummy variable π/2 − x may as well be rewritten as just x, Eq. (2.327) transforms to
– thus unfolding an expression for the sine of a sum of two arguments based on sine and cosine of either argument. Finally, one may rewrite Eq. (2.328) as
again with the aid of Eqs. (2.295) and (2.296); since −y may be exchanged with y in both sides for being a dummy variable, Eq. (2.329) eventually yields
– so the difference between the two cross products of sine and cosine of arguments x and y allows generation of sine of the corresponding difference.
In view of the definition of tangent conveyed by Eq. (2.299), one may write
(2.331)
following insertion of Eqs. (2.321) and (2.330); division of both numerator and denominator by cos x and cos y gives rise to
(2.332)
where insertion of Eq. (2.299) allows simplification to
By the same token, ordered division of Eq. (2.328) by Eq. (2.325) generates
(2.334)
where division of both numerator and denominator of the right‐hand side simultaneously by cos x and cos y unfolds
Eq. (2.335) becomes
after taking Eq. (2.299) into account. Equations (2.333) and (2.336) permit calculation of the tangent of an algebraic sum of arguments, knowing the tangents of the individual arguments.
On the other hand, one may take reciprocals of both sides of Eq. (2.333) to get
(2.337)
with division of both numerator and denominator of the right‐hand side by tan x and tan y yielding
the definition of cotangent as per Eq. (2.304) allows reformulation of Eq. (2.338) to
The same rationale may be applied to Eq. (2.336), viz.
(2.340)
with division of both numerator and denominator of the right‐hand side by tan x tan y unfolding
recalling again the definition of cotangent as reciprocal of tangent, i.e. Eq. (2.304), it is possible to transform Eq. (2.341) to
Equations (2.339) and (2.342) accordingly convey a tool for the calculation of cotan{x ± y} knowing solely cotan x and cotan y.
Upon ordered addition of Eqs. (2.321) and (2.325), one gets
whereas ordered subtraction thereof gives rise to
by the same token, ordered addition of Eqs. (2.328) and (2.330) generates
and ordered subtraction unfolds
One may now define two auxiliary variables, X and Y, according to
and
respectively; ordered addition of Eqs. (2.347) and (2.348) yields
(2.349)
or else
upon isolation of x – whereas ordered subtraction of Eq. (2.348) from Eq. (2.347) leads to
(2.351)
which in turn conveys
when solved for y. Insertion of Eqs. (2.347), (2.348), (2.350), and (2.352) transforms Eq. (2.343) to
and Eq. (2.344) likewise supports
(2.354)
one may similarly transform Eq. (2.345) to
(2.355)
while Eq. (2.346) yields
Equations (2.353)–(2.356) may be useful whenever logarithms of their left‐hand sides are to be handled, because there is no formula to calculate the logarithm of an algebraic sum – unlike happens with the logarithm of a product, see Eq. (2.20).
In attempts to generate further useful relationships involving transformation of arguments of trigonometric functions, it is convenient to resort to the complex domain – and recall that complex numbers, say z1 and z2, may be defined by Cartesian (or rectangular) coordinates as
and
(2.358)
with denoting imaginary unit, besides ai ’s and bi ’s denoting real numbers. Since zi is represented by two coordinates in a plane – ai as abscissa (on the real axis) and bi as ordinate (on the imaginary axis), one may denote as θi the angle of vector of coordinates (ai ,bi) with the real axis, and by ρi its length; hence, Eq. (2.357) may appear alternatively as
and
respectively. Note that
(2.361)
and
(2.362)
were used as defining relationships, in agreement with Eqs. (2.288) and (2.290), complemented by Fig. 2.10 a; and after replacing ‖ u ‖ by ρi, by ai, and by bi . The product of two complex numbers reads
in view of Eqs. (2.359) and (2.360), or else
after lumping ρ1 and ρ2, and applying the distributive property to the product of sums of trigonometric functions; since ι2 = −1 by definition, Eq. (2.364) becomes
along with convenient factoring out of ι. Insertion of Eqs. (2.325) and (2.328) supports transformation of Eq. (2.365) to
in the case of n complex numbers, Eq. (2.366) readily generalizes to
via consecutive application of the transformation of Eq. (2.363) to Eq. (2.366). Should, in addition, ρ1 = ρ2 = ⋯ = ρn = ρ and θ1 = θ2 = ⋯ = θn = θ, then Eq. (2.367) degenerates to
– in view of the functional form conveyed by either Eq. (2.359) or Eq. (2.360), and the definition of power; if ρ is further set equal to unity, then Eq. (2.368) simplifies to
usually known as Moivre's formula – and valid for any positive or negative integer n (as well as for rational numbers). For instance, Eq. (2.369) yields
in the case of n = −1, which breaks down to merely
in view of Eqs. (2.295) and (2.296) – and with z defined
combination of Eqs. (2.370) and (2.371) obviously looks like
Ordered addition of Eqs. (2.371) and (2.372) produces
(2.374)
that may be solved for cos θ as
if Eq. (2.371) is subtracted from Eq. (2.372), then one gets
(2.376)
which gives rise to
after isolation of sin θ. By the same token, one gets
after raising both sides of Eq. (2.372) to the nth power, or else
once reciprocals are taken of both sides; Eq. (2.379) may then be rewritten as
(2.380)
given the rule of composition of powers – where combination with Eq. (2.369) yields
together with Eq. (2.373) upon replacement of θ by nθ. Since Eq. (2.378) may be rewritten as
as per Eqs. (2.369) and (2.372), one concludes that
following ordered addition of Eqs. (2.381) and (2.382), together with cancelation of symmetrical terms afterward; by the same token, ordered subtraction of Eq. (2.381) from Eq. (2.382) generates
In view of Eq. (2.375), one may calculate the power of a cosine via
and Eq. (2.377) similarly supports
after retrieving Newton’s binomial as per Eq. (2.236), it is possible to reformulate Eq. (2.385) to
(2.387)
where the powers of z and of its reciprocal may be lumped to yield
If the exponent of the cosine function is an even integer, say, 2n, then Eq. (2.388) can be redone to
after replacement of n by 2n as upper limit, and concomitant replacement of i by 2i as counting variable of the summation – with subsequent splitting of the said summation, so as to make the median term appear explicitly. At this stage, it is convenient to revisit Eq. (2.240) and realize that
(2.390)
following straightforward algebraic manipulation; in other words,
– i.e. the row entries of Pascal’s triangle are symmetrical relative to its median (see Table 2.1). On the other hand, one may introduce a new counting variable satisfying
so that the second summation in Eq. (2.389) can be algebraically converted to
In view of Eq. (2.391), one may reformulate Eq. (2.393) to
(2.394)
– which, in turn, supports conversion of Eq. (2.389) to
(2.395)
upon lumping of summations for sharing the same lower and upper boundaries, coupled with factoring out of in their kernel; one may now replace dummy variable 2j by merely i, and factor out 2 in the exponents of z and 1/z to get
along with definition of a composite power. Equation (2.383) may finally be invoked to rewrite Eq. (2.396) as
after having the original n replaced by 2(n − i), as well as 2 taken off the outstanding summation; θ was also relabeled as x, provided that rad is used.
If the exponent of cosine in Eq. (2.388) is an odd integer, say, 2n+1, then one gets
upon replacement of n by 2n + 1, and accordingly of i by 2i + 1; the summation was meanwhile rewritten as two consecutive summations, while z in the first summation gave the floor to its reciprocal at the expense of taking the negative of the exponent. A new change of variable, viz.
– inspired on Eq. (2.392), proves useful to transform the second summation in Eq. (2.398) to
where condensation of terms alike meanwhile took place; Eq. (2.391) may then be used to transform Eq. (2.400) to
– while insertion of Eq. (2.401) in Eq. (2.398) further leads to
with summations pooled together on the basis of their similarity, and factored out afterward. Since 2j + 1 is a dummy (counting) variable, it may be swapped for just i, thus allowing reformulation of Eq. (2.402) to
where 2 was meanwhile factored out in the exponents; in view of Eq. (2.383) again – after exchanging exponent n by 2(n − i) + 1, one finds
as alternative version of Eq. (2.403) – also with the aid of 22n being identical to the nth power of 22. Since 2 can cancel out between numerator and denominator, Eq. (2.404) becomes simply
after renaming angle θ to x (expressed in rad); all in all, an integer power of the cosine of an angle x may be expressed as a (finite) sum of cosines of integer multiples of x – see Eqs. (2.397) and (2.405).
With regard to the power of any sine, Eq. (2.386) may be similarly revisited with the aid of Newton’s binomial, labeled as Eq. (2.236), to get
(2.406)
lumping of the powers of z and 1/z supports transformation to
Should the exponent of sine be an even integer, Eq. (2.407) transforms to
once n and i are replaced by 2n and 2i, respectively – where terms were deliberately grouped according to 2i < n, 2i = n and 2i > n, and variable z swapped for 1/z at the expense of a minus sign in the original exponent; variable 2j as per Eq. (2.392) may now be retrieved to rewrite the second summation as
after taking Eq. (2.391) into account and performing elementary algebraic rearrangement. Insertion of Eq. (2.409), together with realization that (−1)2n−2j = (−1)2n (−1)−2j = (−1)−2j = (−1)2j (since 2n and 2j are even integers) and z0 = 1, allow transformation of Eq. (2.408) to
with convenient factoring out of , after having lumped the two summations; upon replacement of 2j by i for being a dummy counting variable, factoring out 2 in the exponents afterward, and splitting the power in denominator, Eq. (2.410) becomes
Equation (2.383) may again be invoked to transform Eq. (2.411) to
(2.412)
while the denominator was rewritten as a product of composite powers – where ι2 = −1 can be used to generate
(2.413)
cancelation of (−1)n between numerator and denominator, and factoring out of 2 in the former unfold
since (−1)i−n coincides with (−1)n−i and x is more generally used as argument than angle θ (as long as rad is employed as units).
In the case of an odd exponent, Eq. (2.407) may be rephrased as
based on change of n to 2n + 1, and likewise of i to 2i + 1, complemented by splitting of the summation in two halves and rearrangement of exponents wherever appropriate; recalling Eq. (2.399), one obtains
as alternative form for the second summation in Eq. (2.415), where condensation of terms alike meanwhile took place. In view of Eq. (2.391) pertaining to the symmetry of binomial coefficients, one may proceed to
(2.417)
as new version of Eq. (2.416) – which may be inserted in Eq. (2.415) to generate
where (−1)2n = (−12)n = 1n = 1 was taken into account after factoring out, (−1)−2j was rewritten as (−1)(−1)−2j−1 = −(−1)−(2j + 1), and was in turn factored out; once 2j + 1 is replaced by i as (dummy) variable in the outstanding summation and ι2 is replaced by −1, Eq. (2.418) simplifies to
along with rewriting of exponent 2n + 1 − 2i as 2(n − i) + 1 and 22n+1 as twice 22n, as well as lumping of powers of −1 between numerator and denominator. After retrieving Eq. (2.384) and replacing n by 2(n − i) + 1, one may reformulate Eq. (2.419) to
(2.420)
in view of (−1)i−n = 1/(−1)i−n = (−1)n−i and 22n = (22)n, which may instead look like
as more usual form – using x (expressed in rad) as independent variable rather than θ; Eqs. (2.414) and (2.421) accordingly permit calculation of an (integer) power of sine of any argument as a linear combination of cosines or sines of multiples of said argument.
The converse problem of expressing sines and cosines of nθ in terms of powers of sin θ and cos θ may also be solved via de Moivre’s theorem; one should accordingly retrieve Eq. (2.369), and expand its left‐hand side via Newton’s binomial formula as
– where advantage was meanwhile taken of ι2 = −1, ι3 = −ι, ι4 = 1, and Eq. (2.236). Following inspection of the forms of the terms in the right‐hand side of Eq. (2.422), one realizes that ι may be factored out to get
(2.423)
a more condensed notation is, however, possible according to
Equation (2.424) may be rewritten as
in the case of an even multiple of θ, materialized via replacement by 2n; and alternatively
when said multiple is odd, i.e. consubstantiated in 2n + 1. Note that no need exists here to change also the form of the counting variable, because no upper limit for the summation was (deliberately) provided in Eq. (2.424) – unlike happened with Eqs. (2.388) and (2.407). For consistency between the linear expression on ι, the coefficients of ι‐dependent and ‐independent terms in both sides of Eq. (2.425) must match – so one may write
(2.427)
complemented with
(2.428)
– based specifically on Eq. (2.425), after expressing 2n − (2j + 1) as 2(n − j) − 1; and similarly
(2.429)
coupled with
(2.430)
– stemming from Eq. (2.426), with convenient factoring out of 2 in both exponents. Note, in either case, the need for a linear combination of cross powers of sine and cosine – with cosine always requiring n + 1 terms, while sine requires n terms for an even multiple but n + 1 terms for an odd multiple; and the (preferred) use of x as argument rather than θ, thus calling for rad as units.
