The student should now have no difficulty in demonstrating the following relations by completing the construction indicated in the margin and attending to the general definitions §§ 20, 27. Some of these relations have already been given or explained. a a a - sin (90° — a); tan (90° - a); cosec a = sec (90° — a). Third Quadrant. = sin a; = cos α = sin (90° -- (12) (13) tan (180° + a) cot (180° + a) = sec (180° + a) cosec (180° + a) sin (360° Fourth Quadrant. α) cos a = sin (90° tan a; = cot α = tan (90° — a); = sec a; = cosec α=- sec (90° — a). may equally express the six functions of all angles in terms of the six functions of angles not greater than 45°. Let y represent any angle not greater than 45°. We may then represent Any angle from 0° to 45° by y; Then, in addition to the relations (12), (13), and (14), which will remain true when we write y instead of a, we shall have the following, which the student should prove. To do this let the student suppose that in the diagram § 26 angle XOM=y, and let him construct the six functions for angles of 90°+y, 270° - y, etc., and compare the lines representing them with the lines on the diagram of § 26. The set corresponding to the first quadrant are already given in (10) and (11). Among the preceding forms of this chapter, the following are of especially frequent application: 1. Express the six functions of the following angles in terms of the three functions sine, tangent, and secant of angles less than 90°: 2. The following table shows the values of four of the functions for every 10° of the first 40° to two places of decimals. By means of these values extend the table to 360°, showing the values of all four functions for each angle: 3. Demonstrate the relations (18) by drawing a diagram showing an arbitrary angle a and an angle 90° greater and less, with the lines representing the sines and cosines. 4. What relations subsist between the following pairs of functions? 30. Special values of trigonometric functions. If angle XOM = 45°, we shall also have OMP = 45°, and therefore Again, whence 2 (b) OT= √0X2 + XT2 = √20X; sec XOM= sec 45° = √2. Next, let angle XOM = 30°. Make angle XOM' = XOM = 30°. The triangles M'OM' and TOT' then have each of their angles 60°, and so are equilateral. Therefore MP = } MM' = } OM = † OX. Τ' (e) (f) (g) Functions of 18°. It is shown in geometry that if the radius of a circle be divided in extreme and mean ratio, the greater segment will be the chord of 36°; that is, twice the sine of 18°. Putting 1 for the radius and r for the greater segment, the condition that the division shall be in extreme and mean ratio is 1:r::r: 1 r, or, equating the product of the means to that of the extremes, r2 = 1 − r. The solution of this quadratic equation. gives The positive root is the only one we want. Hence We then find Hence cos 18° radius- sin' 18° 1 sin' 18°. = - 31. Angles corresponding to given trigonometric functions. When the value of a trigonometric function is given and the angle is required, there are always two solutions to the problem. The Sine. It has already been shown that two supplementary angles have the same sine. Hence if a is an angle corresponding to a given sine, 180° - a will be another angle equally corresponding to it. We may also say that if 90° - ẞ be an angle corresponding to any sine, 90° +ẞ will also correspond to it, because the sum of these two angles is 180°. The same statement applies to the angles 270° ẞ and 270° + B. Unless there is some restriction upon the angle to be chosen, we cannot decide which angle to take. The most common restriction is that the angle must be between the limits - 90° and +90°, or must be in either the first or fourth quadrant. There will then be between these limits one angle and only one for a given sine. Since the measurements of latitude on the surface of the earth are restricted between limits 90° and +90°, the latitude of a place is completely fixed by its sine. The Cosine. The construction of the cosine shows that it has equal values for positive and negative angles. Hence if a be an angle corresponding to a given cosine, a or 360°- a will equally correspond to it. - |