By F. H. C. Marriott (Auth.)
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Extra info for Basic Mathematics for the Biological and Social Sciences
It is a remarkable fact that the different types of curve that can be represented by equation (2) are just the same as the types of curve in which the plane can cut the cone. First of all, if the plane is at right angles to the axis of the cone, the section is a circle. If it is displaced from this position, the circle is distorted—it is stretched in one direction. The resulting curve is called an ellipse (Fig. 9). This is familiar as the curve that represents a circle seen in perspective. $ If a circular tree-trunk is cut at an angle, the section is an ellipse.
If 0 becomes negative, sin 0 at once becomes negative, whereas cos 0 remains positive while 0 > 0 > — π/2. In fact, if 0 is negative, the position of P is the reflection in OX of the corresponding position for 0 positive (it is shown as P' in Fig. 1) so that cos(—0) = cos 0, sin(—0) = —sin 0. It is easy to check, from Fig. 1, the following rules: sin(0 ± 7T) = —sin 0, cos(0 ± π) = —cos 0; sin(0 ± 2&π) = sin 0, cos(0 ± 2&π) = cos 0 when & is any integer; sin(—0) = —sin 0, cos(—0) = cos 0. (1) These rules make it easy to work out the value of sin 0 or cos 0 for any value of 0.
A > c). ] 9. ) |(i) r 2 = Aaz, z = a + br cos 0. (r - cf cr , z2 b2 Λ , (ii) r = kz, z = r(a cos 0 + 6 sin 0) + c. 10. ) (i) r = αθ. (ii) r =
Basic Mathematics for the Biological and Social Sciences by F. H. C. Marriott (Auth.)