By F. H. C. Marriott (Auth.)
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For example, the sine of an angle is unchanged if the angle is increased by 360° or any multiple of 360°. Accordingly if an angle increases steadily, its sine takes the same value at intervals corresponding to 360°. This property can be used to construct mathematical models of seasonal variations, or regular cyclical fluctuations. In Fig. 1, OX and OY are two axes, at right angles, meeting at the centre of a circle of unit radius. P is a point on the circle, PM and PN are the perpendiculars to OX and O Y, and /_POX = 0.
Also, writing A and B for 0 + φ and 0 — φ sin ^4 + sin B = 2 sin cos sin Λ — sin B = 2 sin 4 1 OO cos ^4 + cos B = 2 cos A cos + A B cos ~ B ^ . A + B . A —B cos ^4 — cos B = — 2 sin sin . /11X (11) Finally sin 2 0 = 2 sin 0 cos 0 cos 20 = cos20 — sin20 = 2 cos20 - 1 = 1 - 2 sin20 (12) and, by dividing the formula for sin(0 + φ) by that for cos(0 + φ)9 ——— r -. (13) 1 — tan 0 tan φ All these formulae need not be memorized. The relationships in (1) should be known, and the definitions in (2). The forms of Pythagoras' theorem in (3)-(5) should be familiar, but the first one leads immediately to the other two.
A parabola. follow a straight-line relationship. The equation^ = a + bx — ex2 can be used to represent the relationship between the yield of a crop and the dressing of fertilizer, between the weight of an animal and time, between the income of a business and expenditure on advertising, and many other practical situations. In all these cases, while the equation represents a parabola, the relationship is applicable over quite a restricted range of values of x. Generally the equation begins to break down before y reaches its 30 MATHEMATICS FOR BIOLOGICAL AND SOCIAL SCIENCES maximum value and starts to decrease.