By F. H. C. Marriott (Auth.)

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1, by Pythagoras' theorem, PM2 + OM2 = OP2= 1. sin20 + cos20 = l. Hence (3) 2 Dividing by cos 0 and rearranging: sec20 - t a n 2 0 = I. (4) 2 Dividing (3) by sin 0 and rearranging: cosec20 — cot20 = l. (5) 46 MATHEMATICS FOR BIOLOGICAL AND SOCIAL SCIENCES These rules are thus simply different forms of Pythagoras' theorem. It is not quite so easy to prove the formulae for the sine and cosine of sums of angles. In Fig. 2, OP makes an angle 0 + φ with OX, and PL is the perpendicular from P to OZ9 which makes an angle 0 with OX.

FIG. 2. The summation formulae. See text. Now OL = cos φ, PL = sin φ, and Z_MPL = 0. The projection of OP on OX = the projection of OL on OX — the projection of PL on OX. Hence cos(0 + φ) = cos 0 cos φ — sin 0 sin φ. (6) Similarly from the projections on O Y, sin(0 + φ) = sin 0 cos φ + cos 0 sin φ. (7) These results have been proved only for an acute angle 0 + φ. They are in fact true for any angles 0 and φ. This can be proved using equations (1). Changing the sign of φ gives: sin(0 — φ) = sin 0 cos φ — cos 0 sin <£, (8) cos(0 — Φ) = cos 0 cos <£ + sin 0 sin <£.

The distances rA and rB are sometimes used as bipolar coordinates,] 15. The shape of a tree-trunk is sometimes represented in cylindrical polars by the equation r = a + bz + cz2. Describe the surface, and explain how the values of a, b and c are related to the height, diameter and shape of the trunk. 1. Definitions In elementary trigonometry, the sine and cosine of an angle, and the other related functions, are defined as the ratios of the sides of a right-angled triangle. This is the definition that is normally needed in surveying and map-making, but does not bring out the property that is most important in economic and biological applications.