compound angle
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A Geometric View of the Trigonomentry Compound Angle FormulaeAuthor(s): Stewart FowlieSource: Mathematics in School, Vol. 33, No. 3 (May, 2004), p. 15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215696 .
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A
5Ei571i7RIC VIEW
OF THE TRIGONOMETRY COMPOUND
AMGLE FOR17UL5E
by Stewart Fowlie
David Pagni's article in May 2003 presented a geometric view of the six trigonometric functions, each as a length rather than as a ratio, and in a similar way I give diagrams in which the formulae for sin(X + Y) and cos(X + Y) are represented.
With sinA and cosA defined as in (i), where A can be any angle, it is clear that sin(180+ - A) = sinA, but cos (1800 - A) = cosA.
In (ii), thought of as a side of triangle ABD, AD = c sin B, but as a side of triangle ACD, AD = b sin C. Thus b sin C =
b c csinB, or sinB - sinC Similarly each of these fractions is
a equal to sinA . This, of course, is the sine formula, and smnA
means that a triangle with sides sinA, sinB and sinC has, as its angles, A, B and C as in (iii). It may be seen in triangle ABD that BD = cosB sinC and DC = cosC sinB. (Also in each triangle that DA = sin B sin C.)
Thus sin(B + C) = sin (1800 - A) = sinA (as remarked above) = BD + DC = cosB sinC + cosC sinB. Rearranging to the customary form gives
sin (X + Y) = sin X cos Y + cos X sin Y.
Diagram (iv) is diagram (iii) repeated with BE drawn cutting AD at H, the orthocentre of triangle ABC. Since from triangle BEC, angle HBD = 900 - C, angle BHD = C. Thus looking at triangle HBD, since BD = cosB sinC, BH = cosB and DH = cosB cosC. Similarly, AH = cosA.
Thus cos(B + C) = cos (1800 - A) = -cosA (as remarked above) = -(AD - HD) = -sinB sinC + cosB cosC. Rearranging to the customary form gives
cos(X + Y) = cosX cosY - sinX sinY.
Formulae for sin(X - Y) and cos(X - Y) may be demonstrated by taking the point C between B and D, making the triangle ABD obtuse angled at C.
Keywords: Trigonometry; Compound angle formulae.
Author Stewart Fowlie, 24 Granton Road, Edinburgh EH5 3QH. e-mail: [email protected]
(i)
(ii)
(iii)
(iv)
sinA
A cosA
A
c b csinB bsinC
B D C
A
sinC sinB sinBsinC
cosBsinC r1 cosCsinB B D C
A
E H
csC cosBcosC
cosBsinC I B D C
Mathematics in School, May 2004 The MA web site www.m-a.org.uk 15
This content downloaded from 132.235.61.22 on Sat, 5 Oct 2013 18:11:15 PMAll use subject to JSTOR Terms and Conditions