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2.3.3 Fundamental Theorem of Trigonometry
ОглавлениеConsider a right triangle, i.e. a triangle containing a right angle – as depicted in Fig. 2.11a. In Euclidean geometry, a fundamental relationship exists encompassing the three sides of any right triangle – namely, the square of the hypotenuse (or side opposite to the right angle), of length c, equals the sum of the squares of the other two sides, of lengths a and b. In other words,
which is classically known as Pythagorean equation – in honor to ancient Greek mathematician Pythagoras (570–495 BCE), historically credited for its first (recorded) proof.
Figure 2.11 Illustration of Pythagoras’ theorem as (a) graphical statement, and graphical proof based on (b) combination of simple polygons or (c) relationships between similar triangles.
Despite the 400+ distinct proofs available, one may to advantage take four copies of a right triangle with sides a, b, and c – arranged inside a square with side c, as outlined in Fig. 2.11 b; the triangles share their area, ab/2 (a formula to be derived in due course), and the smaller square has side b − a. The area c2 (also to be derived) of the larger square may thus be given by
(2.432)
where (b − a)2 represents the area of the smaller square; expansion of the square of the binomial in the right‐hand side as per Eq. (2.238), followed by lumping of constants between numerator and denominator in the last term unfold
(2.433)
– whereas cancelation of symmetrical terms immediately retrieves Eq. (2.431).
One may alternatively resort to the proportionality of the sides of two similar triangles; let [ABC] accordingly represent a right triangle, with the right angle located at C – as per Fig. 2.11c. After drawing an altitude from point C toward side [AB], and denoting its intersection with said side as D, one ends up with point D dividing the hypotenuse [AB] into segments [AD] and [BD]. The new triangle [ACD] is similar to triangle [ABC], because they both have a right angle – at C in [ABC] (by hypothesis) and at D in [ACD] (by definition of altitude), and they share angle θ at A; therefore, the remaining angles must be identical, since the angles of a triangle always add up to π rad (see also proof at a later stage). By the same token, triangle [BCD] is similar to [ABC] – since they share the angle at B, and they both have a right angle, i.e. ∠BDC and ∠ACB, respectively. Remember that similarity of triangles enforces equality of ratios of the corresponding sides, i.e.
relating length of sides in [BCD] and [ABC] opposed to right angle as left‐hand side, and relating length of sides also in [BCD] and [ABC] opposed to angle of amplitude θ (note the mutually perpendicular sides of said angles) as right‐hand side; coupled with
relating length of sides in triangles [ACD] and [ABC] opposed to the right angle as left‐hand side, and relating length of sides also in [ACD] and [ABC] opposed to angle of amplitude π − π/2 − θ = π/2 − θ as right‐hand side. Equation (2.434) may be rewritten as
upon elimination of denominators – and a similar result, viz.
can be produced from Eq. (2.435); ordered addition of Eqs. (2.436) and (2.437) unfolds
(2.438)
where may, in turn, be factored out to give
Since [BD] and [AD] are consecutive straight segments, Eq. (2.439) is equivalent to
(2.440)
this relationship coincides with Eq. (2.431), because , , and , i.e. the sides of a right triangle, as per comparative inspection of Fig. 2.11a and c.
When a, b, and c in Eq. (2.431) represent sides [OA], [AB], and [OB] (or u, for that matter), respectively, in Fig. 2.10a, with lengths , , and , one may resort to Eqs. (2.289) and (2.291) to reformulate Eq. (2.431) as
(2.441)
or else
after taking Eq. (2.287) into account and replacing θ by x (as usual); Eq. (2.442) is known as fundamental theorem of trigonometry.
Figure 2.12 Graphical representation of generic triangle [ABC] – with indication of corners A, B, and C, lengths of opposite sides a (corresponding to [BC]), b (corresponding to [AC]), and c (corresponding to [AB]), and angles of adjacent sides (a, b, c) α (formed by [AB] and [AC]), (a,b) β (formed by [AB] and [BC]), and (a,b) γ (formed by [AC] and [BC]) – after drawing an altitude (b) from A to [BC], B to [AC], or C to [AB], or (c) from the center, O, of circumcircle (ABC) to point D on [BC].
The Pythagorean theorem is a special case of a more general theorem relating the lengths of the sides of any triangle (not necessarily containing a right angle), viz.
– which degenerates to Eq. (2.431) when γ (i.e. the angle formed by sides of length a and b) equals π/2, since the corresponding cosine is nil); this is usually known as cosine formula (or cosine rule), and abides to the nomenclature in Fig. 2.12a. Equation (2.443) is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known; it was explicitly stated only in the fifteenth century, by Arab mathematician Jamshid al‐Kashi. By changing which sides of the triangle are denoted as a, b, and c, Eq. (2.443) may appear as
(2.444)
or else
(2.445)
– encompassing angles α and β, respectively. To prove the validity of Eq. (2.443), one may to advantage drop the perpendicular from corner A onto side [BC], as illustrated in Fig. 2.12b – so the definition of cosine as per Eq. (2.288) allows one to write
where the first term in the right‐hand side represents the length of the portion of [BC] closer to B, and the second term represents the remainder of [BC] closer to C, upon multiplication of both sides by a, Eq. (2.446) becomes
A similar rationale may be followed with regard to the perpendicular from corner B to side [AC] – see Fig. 2.12b, where again the definition of cosine supports
here the first term in the right‐hand side represents the length of the portion of [AC] closer to A, and the second term represents the length of the remainder of [AC] closer to C. Multiplication of both sides by b then converts Eq. (2.448) to
By the same token, a perpendicular can be dropped from corner C to side [AB] in Fig. 2.12b to yield
(2.450)
where the first term in the right‐hand side represents the length of the portion of [AB] closer to A, while the second term represents the length of the remainder of [AB] closer to B; c may then multiply both sides to produce
Ordered addition of Eqs. (2.447) and (2.449) gives rise to
(2.452)
or else
after condensing terms alike; ordered subtraction of Eq. (2.451) from Eq. (2.453) generates
(2.454)
which breaks down to
(2.455)
upon cancelation of symmetrical terms – and which recovers Eq. (2.443), after swapping 2ab cos γ and c2 between sides.
Another important relationship in trigonometry relates the lengths of the sides of a triangle of any shape to the sines of its angles, according to
it is classically referred to as law of sines, or sine formula – and a, b, and c denote lengths of the sides opposing angles α, β, and γ, respectively, while R denotes radius of the circumcircle to the triangle. To derive the first two equalities in Eq. (2.456), one should recall the rule of calculation of the area of a triangle, S – as one half of the product of its base by its height (to be proven later); inspection of Fig. 2.12b indicates that this can be done in three different ways, i.e.
using [BC] as base and [AC] as hypotenuse,
using [AC] as base and [AB] as hypotenuse, or
using [AB] as base and [BC] as hypotenuse – in all cases at the expense of the graphical interpretation of sine conveyed by Eq. (2.290). Since Eqs. (2.457)–(2.459) share their left‐hand side, one may lump them as
(2.460)
whereas a further multiplication of all sides by 2/abc permits simplification to
the first two equalities in Eq. (2.461) degenerate to the corresponding equalities in Eq. (2.456), after taking reciprocals of all three sides. To prove the last equality of Eq. (2.456), one should redraw triangle [ABC] together with the corresponding circumscribed circle – as done in Fig. 2.12 c; furthermore, an altitude may be drawn from the center of the circle, O, toward side [BC], with intercept denoted as D. Since sides [OB] and [OC] of auxiliary triangle [OBC] are identical for coinciding with radius R of the circle, the altitude [OD] of this isosceles triangle splits in half the angle formed by [OB] and [OC], as well as segment [BC] of length a; therefore, angle ∠BOD is equal to angle ∠COD, and to half the angle ∠BOC – and thus also equal to angle ∠BAC of amplitude α, because it subtends side [BC] with corner A lying on the circle. By definition of sine as ratio of length of opposite leg to that of hypotenuse, one realizes that
(2.462)
pertaining to ∠BOD, or equivalently
after revisiting Eq. (2.458) as
following isolation of sin α, one may eliminate sin α between Eqs. (2.463) and (2.464) to get
(2.465)
– which degenerates to
when solved for the reciprocal of 2R. Insertion of Eq. (2.466) transforms Eq. (2.461) to
(2.467)
– so Eq. (2.456) will be retrieved in full, after reciprocals are taken of all four sides.
Once in possession of Eq. (2.442), one may produce an alias through division of both sides by sin2 x, viz.
(2.468)
which is equivalent to
in view of Eqs. (2.304) and (2.314); if division of both sides is performed by cos2 x instead, then Eq. (2.442) becomes
(2.470)
– which may appear as
at the expense of Eqs. (2.299) and (2.309). Both Eqs. (2.469) and (2.471) may be useful, namely when changing variables throughout integration of some types of functions.
Figure 2.13 Variation, with their argument x, of major inverse trigonometric functions, viz. (a) inverse sine (sin−1) and cosine (cos−1), and (b) inverse tangent (tan−1) and cotangent (cotan−1).
