EXAMPLES.-VIII. Express in circular measure the following angles : (1) 60o. (2) 22°30'. (3) 11° 15'. (5) 315o. (6) 24°. 13'. (9) The angles of an equilateral triangle. (10) The angles of an isosceles right-angled triangle. 42. If the circular measure of an angle be given to find the number of degrees which it contains. Let be the given circular measure. Hence we obtain the following rule: If an angle be expressed in circular measure, multiply the measure by 180, divide the result by 7, and you obtain the measure of the angle in degrees. Ex. Express in degrees the angle whose circular measure Express in degrees, &c. the angles whose circular measures 43. Similar rules will hold with respect to the equations for connecting the centesimal and the circular systems, 200 being put in the place of 180: thus circular measure of an angle containing G grades = G. number of grades in the angle whose circular measure is 0 = 0. π 200 200 Express in grades, &c. the angles whose circular measures 44. We shall now give a set of Miscellaneous Examples to illustrate the principles explained in this and the two preceding Chapters. EXAMPLES.-XII. 1. If the unit of angular measurement be 5°, what is the measure of 2210? 2. If an angle of 421° be represented by 10, what is the unit of measurement ? 3. An angle referred to different units has measures in the ratio 8 to 5; one unit is 2o, what is the other? Express each unit in terms of the other." 4. An angle referred to different units has measures in the ratio 7 to 6; one unit is 3o, what is the other? Express each unit in terms of the other. 5. If half a right angle be taken as the unit of angular measurement, what is the measure of an angle of 42o ? 6. Compare the angles 13°. 13'. 48" and 148.7. 7. If D be the number of degrees in any angle and G the number of grades, shew that G=D+ + D. 9 8. An equilateral triangle is divided into two triangles by a line bisecting one side; express the angles of these two triangles in degrees and grades respectively. 9. It being given that the angle subtended by an arc equal to the radius is 57°-29577, find the ratio of the circumference of a circle to its diameter. : 10. Two angles of a triangle are in magnitude as 2 3. If the third angle be a right angle, express the angles of this triangle in each of the three systems of measurement. 11. Two straight lines drawn from the centre of a circle contain an angle subtended by an arc which is to the whole circumference as 13: 27; express this angle in degrees. 12. An arc of a circle is to the whole circumference as 17:54; express in grades the angle which the arc subtends at the centre of the circle. 13. If the angles of a triangle are in Arithmetical Progression, shew that one of them is 60°. 14. Determine in grades the magnitude of the angle subtended by an arc two feet long at the centre of a circle whose radius is 18 inches. 15. Compare the measures of 23°. 5′ when expressed in centesimal and circular measure, taking = 22 16. If a degree be represented by π, what is the unit of angular measurement ? 17. What is the circular measure of 118o. 30` if π = ? 355 113 18. Reduce 398-012 to degrees, minutes and seconds. 19. If the numerical value of an angle measured by the 20. The whole circumference of one circle is just long enough to subtend an angle of one grade at the centre of another circle: what part of the latter circumference will subtend an angle of one degree at the centre of the former circle? 21. Taking 4 right angles as the unit, what number will represent 1°, 18, 1° respectively? If there be m English minutes in an angle, find the number of French seconds in the same angle. 23. What is the number of grades in the unit of circular 22 measure, if π= ? 24. What fraction of a right angle must be the unit in order that an angle of 5°. 33'. 20" may be represented by 5? 25. If half a right angle be the unit of angular measurement, express the angles whose measures are (1) in degrees, (2) in units of circular measure. 26. What must be the unit angle if the sum of the measures of a degree and a grade is 1? 27. If the unit be an angle subtended at the centre of a circle by an arc three times as large as the radius, what number will represent an angle of 45° ? 28. If there be three angles in arithmetical progression, and the number of grades in the greatest be equal to the number of degrees in the sum of the other two, the angles are as 11; 19: 27. Express in degrees: 29. (1) The angle of a regular hexagon. (2) The angle of a regular pentagon. 30. Express in grades: (1) The angle of a regular pentagon. (1) The angle of an equilateral triangle. 32. Prove that = 115%. 47' nearly. 180° 33. Find the circular measure of the angle of a regular polygon of n sides. 34. The radius of a circle is 18 feet, find the length of an arc which subtends an angle 10° at the centre. 35. The three angles of a triangle are in arithmetical progression, and the number of grades in the least: the number of degrees in the greatest :: 2 : 9. Find the angles. 36. How many degrees are there in the angle whose circular measure is 2.5 ? 37. One angle of a triangle is 2 in circular measure, and another is 20°: find the number of grades in the third. 38. An arc of a circle, whose radius is 7 inches, subtends an angle of 15o. 39′. 7′′; what angle will an arc of the same length subtend in a circle whose radius is 2 inches? 39. The earth being supposed a sphere of which the diameter is 7980 miles, find the length of 1° of the meridian. 40. The angles in one regular polygon are twice as many as those in another polygon, and an angle of the former : an angle of the latter 3: 2. Find the number of the sides in each. |