CHAPTER II. TRIGONOMETRICAL RATIOS. 9. DEFINITION. Ratio is the relation which one quantity X bears to another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other. To find what multiple or part A is of B we divide A hence the ratio of A to B may be measured by the by B; fraction A B In order to compare two quantities they must be expressed in terms of the same unit. Thus the ratio of 2 yards to 2 × 3 × 12 8 27 inches is measured by the fraction or 27 3* OBS. Since a ratio expresses the number of times that one quantity contains another, every ratio is a numerical quantity. 10. DEFINITION. If the ratio of any two quantities can be expressed exactly by the ratio of two integers the quantities are said to be commensurable; otherwise, they are said to be incommensurable. For instance, the quantities 8 and 5 are commensurable, while the quantities √2 and 3 are incommensurable. But by finding the numerical value of √2 we may express the value of the ratio 2:3 by the ratio of two commensurable quantities to any required degree of approximation. Thus to 5 decimal places √2=1·41421, and therefore to the same degree of approximation √2:3=1·41421 : 3=141421 : 300000. Similarly, for the ratio of any two incommensurable quantities. Trigonometrical Ratios. 11. Let PAQ be any acute angle; in AP one of the boundary lines take a point B and draw BC perpendicular to AQ. Thus a right-angled triangle BAC is formed. With reference to the angle 4 the following definitions are employed. BC opposite side The ratio or AB is called the sine of A. is called the cosine of A. is called the tangent of A. is called the cotangent of A. is called the secant of A. is called the cosecant of A. These six ratios are known as the trigonometrical ratios. It will be shewn later that as long as the angle remains the same the trigonometrical ratios remain the same. [Art. 19.] 12. Instead of writing in full the words sine, cosine, tangent, cotangent, secant, cosecant, abbreviations are adopted. Thus the above definitions may be more conveniently expressed and arranged as follows: In addition to these six ratios, two others, the versed sine and coversed sine are sometimes used; they are written vers A and covers A and are thus defined: vers A=1-cos A, covers A = 1-sin A. 13. In Chapter VIII. the definitions of the trigonometrical ratios will be extended to the case of angles of any magnitude, but for the present we confine our attention to the consideration of acute angles. 14. Although the verbal form of the definitions of the trigonometrical ratios given in Art. 11 may be helpful to the student at first, he will gain no freedom in their use until he is able to write down from the figure any ratio at sight. In the adjoining figure, PQR is a right-angled triangle in which PQ=13, PR=5, QR=12. Since PQ is the greatest side, R is the right angle. The trigonometrical ratios of the angles P and Q may be written down at once; for example, 13 12 PQ 13' cos Q = cosec P= PQ QR = 13 12 R 5 15. It is important to observe that the trigonometrical ratios of an angle are numerical quantities. Each one of them represents the ratio of one length to another, and they must themselves never be regarded as lengths. 16. In every right-angled triangle the hypotenuse is the greatest side; hence from the definitions of Art. 11 it will be seen that those ratios which have the hypotenuse in the denominator can never be greater than unity, while those which have the hypotenuse in the numerator can never be less than unity. Those ratios which do not involve the hypotenuse are not thus restricted in value, for either of the two sides which subtend the acute angles may be the greater. Hence the sine and cosine of an angle can never be greater than 1; the cosecant and secant of an angle can never be less than 1; the tangent and cotangent may have any numerical value. 17. Let ABC be a right-angled triangle having the right angle at A; then by Euc. I. 47, the sq. on BC sum of sqq. on AC and AB, or, more briefly, BC2 AC2+AB2. When we use this latter mode of expression it is understood that the sides AB, AC, BC are expressed in a terms of some common unit, and the above statement may be regarded as a numerical relation connecting the numbers of units of length in the three sides of a right-angled triangle. It is usual to denote the numbers of units of length in the sides opposite the angles A, B, C by the letters a, b, c respectively. Thus in the above figure we have a2=b2+c2, so that if the lengths of two sides of a right-angled triangle are known, this equation will give the length of the third side. Example 1. ABC is a right-angled triangle find Example 2. A ladder 17 ft. long is placed with its foot at a distance of 8 ft. from the wall of a house and just reaches a windowsill. Find the height of the window-sill, and the sine and tangent of the angle which the ladder makes with the wall. Let AC be the ladder, and BC the wall. Let x be the number of feet in BC; then x2=(17)2 — (8)2= (17+8) (17 – 8)=25 × 9 ; 18. The following important proposition depends upon the property of similar triangles proved in Euc. vI. 4. The student who has not read the sixth Book of Euclid should not fail to notice the result arrived at, even if he is unable at this stage to understand the proof. 19. To prove that the trigonometrical ratios remain unaltered so long as the angle remains the same. Let AOP be any acute angle. In OP take any points B and X B P E D, and draw BC and DE perpendicular to OA. Also take any point Fin OP and draw FG at right angles to OP. But the triangles BOC, DOE, FOG are equiangular ; Thus the sine of the angle POA is the same whether it is obtained from the triangle BOC, or from the triangle DOE, or from the triangle FOG. A similar proof holds for each of the other trigonometrical ratios. These ratios are therefore independent of the length of the revolving line and depend only on the magnitude of the angle. |