§ 23. EXTENSION OF FORMULAS [1]-[3] TO ALL ANGLES. The Formulas established for acute angles in § 6 hold true for all angles. Thus, Formula [1], sin2x+cosx=1 1, is universally true; for, whether MP and OM (Fig. 25) are positive or negative, MP2 and OM2 are always positive, and in each quadrant MP2+OM2 = (P2 = 1. are universally true; for they are in harmony with the algebraic signs of the functions, given at the end of § 20; and we have in each quadrant from the similar triangles OMP, OAT, OBS, (Fig. 25) the proportions. AT: OA-MP : OM, MP : 0P=0B : OS, OM : OP=OA : OT, AT: OA=0B : BS, which, by substituting 1 for the radius, and the right names. for the other lines, are easily reduced to the above formulas. Formulas [1]-[3] enable us, from a given value of one function, to find the absolute values of the other five functions, and also the sign of the reciprocal function. But in order to determine the proper signs to be placed before the other four functions, we must know the quadrant to which the angle in question belongs; or the sign of any one of these four functions; for, by (§ 20) it will be seen that the signs. of any two functions that are not reciprocals determine the quadrant to which the angle belongs. EXAMPLE. Given sin x+, and tan x negative; find the values of the other functions. Since sin x is positive, x must be an angle in Quadrants I. or II.; but, since tan x is negative, Quadrant I. is inadmissible. Since the angle is in Quadrant II. the minus sign must be taken, and we have 1. Construct the functions of an angle in Quadrant II. What are their signs? 2. Construct the functions of an angle in Quadrant III. What are their signs? 3. Construct the functions of an angle in Quadrant IV. What are their signs? 4. What are the signs of the functions of the following angles: 340°, 239°, 145°, 400°, 700°, 1200°, 3800° ? 5. How many angles less than 360° have the value of the sine equal to +, and in what quadrants do they lie? 6. How many values less than 720° can the angle x have if cos x+, and in what quadrants do they lie? 2 7. If we take into account only angles less than 180°, how many values can x have if sin x § ? if cos x=? if cos x= -? if tan x= 2 ? if cotx=- -7 ? 8. Within what limits must the angle x lie if cos x=- 3 ? if cotx=4? if sec x 4? if secx=80 ? if csc x —— -3? (if x < 360°). 9. In what quadrant does an angle lie if sin and cosine are both negative? if cosine and tangent are both negative? if the cotangent is positive and the sine negative? 10. Between 0° and 3600° how many angles are there whose sines have the absolute value ? Of these sines how many are positive and how many negative ? 11. In finding cos x by means of the equation cos x= ±√1—sin2x, when must we choose the positive sign and when the negative sign? 12. Given cos x= √; find the other functions when x is an angle in Quadrant II. 13. Given tan x=√3; find the other functions when x is an angle in Quadrant III. 14. Given sec x=+7, and tan x negative; find the other functions of x. 15. Given cotx=-3; find all the possible values of the other functions. 16. What functions of an angle of a triangle may be negative? In what case are they negative? 17. What functions. of an angle of a triangle determine the angle, and what functions fail to do so? 18. Why may cot 360° be considered equal either to +∞o or to ∞ ? 19. Obtain by means of Formulas [1]-[3] the other func 20. Find the values of sin 450°, tan 540°, cos 630°, cot 720°, sin 810°, csc 900°. 21. For what angle in each quadrant are the absolute values of the sine and cosine equal? Compute the values of the following expressions: 22. a sin 0°+b cos 90° — c tan 180°. 23. a cos 90° — b tan 180° + c cot 90°. 24. a sin 90° b cos 360° + (a - b) cos 180°. 25. (a2 — b2) cos 360° — 4 ab sin 270°. § 24. REDUCTION OF FUNCTIONS TO THE FIRST QUADRANT. In a unit circle (Fig. 26) draw two diameters PR and QS B N A M Ꭱ B' FIG 26. A equally inclined to the horizontal diameter AA', or so that the angles AOP, A'OQ, A'OR, and AOS shall be equal. From the points P, Q, R, S let fall perpendiculars to AA'; the four right triangles thus formed, with a common vertex at O, are equal; because they have equal hypotenuses (radii of the circle) and equal acute angles at O. There fore, the perpendiculars PM, QN, RN, SM, are equal. Now these four lines are the sines of the angles AOP, AOQ, AOR, and AOS, respectively. Therefore, in absolute value, sin AOP=sin AOQ= sin AOR=sin AOS. And from § 23 it follows that in absolute value the cosines of these angles are also equal; and likewise the tangents, the cotangents, the secants, and the cosecants.* Hence, for every acute angle (AOP) there is an angle in each of the higher quadrants whose functions, in absolute value, are equal to those of this acute angle. Let ▲ AOP=x, ▲ POB=y; then x+y=90°, and the functions of x are equal to the co-named functions of y (§ 5); and 40Q (in Quadrant II.) 180°-x= 90°+y, AOQ ZAOR (in Quadrant III.)=180°+x=270° — y, ▲ AOS (in Quadrant IV.) =360° — x=270°+y. Hence, prefixing the proper sign (§ 20), we have: * In future, secants, cosecants, versed sines, and coversed sines will be disregarded. Secants and cosecants may be found by [3], versed sines and coversed sines by VII. and VIII., page 5, if wanted, but they are seldom used in computations. REMARK. The tangents and cotangents may be found directly from the figure, or by formula [2]. It is evident from these formulas, 1. The functions of all angles can be reduced to the functions of angles not greater than 45°. 2. If an acute angle be added to or subtracted from 180° or 360°, the functions of the resulting angle are equal in absolute value to the like-named functions of the acute angle; but if an acute angle be added to or subtracted from 90° or 270°, the functions of the resulting angle are equal in absolute value to the co-named functions of the acute angle. 3. A given value of a sine or cosecant determines two supplementary angles, one acute, the other obtuse; a given value of any other function determines only one angle: acute if the value is positive, obtuse if the value is negative. [See functions of · (180° — x).] |