EXAMPLES. 1. Find the number of degrees in the angle whose circular measure is 1 Here A = Х T 90 x 7 = 28° 38' 10'10 22 where 42 is used for . 2. Find the circular measure of the angle 59° 52' 30". Express the angle in degrees and decimals of a degree thus : 60)52.5 59.875 ..0 = 59.8 7 5 T = 180 :(.333 ...) == =1.0453 .. 3. Express, in degrees, the angles whose circular measures NOTE 1. - The student should especially accustom himself to express readily in circular measure an angle which is given in degrees. 40 5. Express in circular measure 3° 12', and find to seconds the angle whose circular measure is .8. 22 Take a = Ans. 45° 49' 5" 1 7 225' 6. One angle of a triangle is 45°, and the circular measure of another is 1.5. Find the third angle in degrees. Ans. 49° 5' 27" 11: (Take NOTE 2. - Questions in which angles are expressed in different systems of measurement are easily solved by expressing each angle in right angles. 7. The sum of the measure of an angle in degrees and twice its measure in radians is 234; find its measure in degrees (n = 22). Let the angle contain x right angles. = 90x. 66 66 66 66 .:. 90 α +πα 234; 163 .. 90 x + 22x ; 7 1 .:: 652x = 163, .. X = :: the angle is of 90° = 221°. 8. The difference between two angles is and their sum 9' is 56°; find the angles in degrees. Ans. 38, 18° ° °. 11. General Measure of an Angle. In Euclidian geometry and in practical applications of trigonometry, angles are generally considered to be less than two right angles; but in the theoretical parts of mathematics, angles are treated as quantities which may be of any magnitude what ever. B Thus, when we are told that an angle is in some particular quadrant, say the second (Art. 5), we know that the position in which the revolving line stops is in the second quadrant. But there is an unlimited number of angles having the same final position, OP. The revolving line OP may pass from OA to OP, not only by describing the P arc ABP, but by moving through a whole revolution plus the arc ABP, or through any number of revolutions plus the arc ABP. For example, the final position of OP may represent geometrically all the following angles : Angle AOP - 130°, or 360° +130°, or 720° +130°, or - 360° +130°, or 720° + 130°, etc. Let A be an angle between 0 and 90°, and let n be any whole number, positive or negative. Then (1) 2n x 180° + A represents algebraically an angle in the first quadrant. (2) 2n x 180o – A represents algebraically an angle in the fourth quadrant. (3) (2n +1) 180° – A represents algebraically an angle in the second quadrant. (4) (2n +1) 180° + A represents algebraically an angle in the third quadrant. In circular measure the corresponding expressions are (1) 2nT+0, (2) 2n7-6, (3) (2n+1)=–0, (4) (2n+1) +0. EXAMPLES. State in which quadrant the revolving line will be after describing the following angles : (1) 120°, (2) 340°, (3) 490', (4) – 100°, 4 12. Complement and Supplement of an Angle or Arc. — The complement of an angle or arc is the remainder obtained by subtracting it from a right angle or 90°. The supplement of an angle or arc is the remainder obtained by subtracting it from two right angles or 180°. Thus, the complement of A is (90° – A). The complement of 190° is (90° — 190°)=-100°. The complement of £r is (5 – *=) == }r. T. The supplement of in is (1 - 1)=1*. EXAMPLES. : 1. If 192 square inches be represented by the number 12, what is the unit of linear measurement ? Ans. 4 inches. 2. If 1000 square inches be represented by the number 40, what is the unit of linear measurement? Ans. 5 inches. 3. If 2000 cubic inches be represented by the number 16, what is the unit of linear measurement ? Ans. 5 inches. 4. The length of an Atlantic cable is 2300 miles and the length of the cable from England to France is 21 miles. Express the length of the first in terms of the second as unit. Ans. 10911 5. Find the measure of a miles when b yards is the unit. 1760a Ans. b 6. The ratio of the area of one field to that of another is 20:1, and the area of the first is half a square mile. Find the number of square yards in the second. Ans. 77440. 7. A certain weight is 3.125 tons. What is its measure in terms of 4 cwt.? Ans. 15.625. Express the following 12 angles in centesimal measure: 8. 42° 15' 18". Ans. 468 95'. 9. 63° 19' 17". 708.35' 70'.98 .... 10. 103° 15' 45". 1148 73' 61".1. 11. 19° 0' 18". 218 11' 66"... 12. 143° 9' 0". 1598 5' 55".5. 13. 300° 15' 58". 3338 62' 90".1234567890. 14. 27° 41' 51", 308.775. 15. 67°.4325. 748.925. 16. 8° 15' 27". 98 17' 50". 17. 97° 5' 15". 1078 87' 50". 18. 16° 14' 19". 185 4' 29"\.... 19. 132° 6'. 1468 77' 77'7. Express the following 11 angles in degrees, minutes, and seconds : Express in circular measure the following angles : 14537 31. 315°, 24° 13'. 10800 27197 32. 95° 20', 12° 5' 4". 270 40500 Ans. T, 143 TT п T. 33. 221°, 1°, 57o.295. 1 radian. 8' 180' 34. 120°, 45°, 270°. 2.09439, 1 ft. , 4' 35. 360°, 3. rt. angles. 27, T. Express in degrees, etc., the angles whose circular meas ures are: |