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36. The units in the three systems when expressed in terms of one common standard, two right angles, stand thus :

the unit in the Sexagesimal Method =

1

180

of two right angles,

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37. It is not usual to assign any distinguishing mark to angles estimated by the Third Method, but for the purpose of stating the relation between the three units in a clear and concise form, we shall use the symbol 1° to express the unit of circular measure.

Then we express the relation between the units thus:

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CHAPTER III.

ON THE METHOD OF CONVERTING THE MEASURES OF ANGLES

FROM ONE TO ANOTHER SYSTEM OF MEASUREMENT.

38. WE proceed to explain the process for converting the measures of angles from each of the three systems of measurement described in Chap. II. to the other two.

39. To convert the measure of an angle expressed in degrees to the corresponding measure in grades.

Let the given angle contain D degrees.

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Hence we obtain the following rule: If an angle be expressed in degrees, multiply the measure in degrees by 10, divide the result by 9, and you obtain the measure of the angle in grades.

Ex. How many grades are contained in the angle 24°. 51'. 45′′? 24°. 51′. 45′′ = 24.8625 degrees

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EXAMPLES.-VI.

Find the number of grades, minutes and seconds in the follow

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40. To convert the measure of an angle expressed in grades

to the corresponding measure in degrees.

Let the given angle contain G grades.

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Hence we obtain the following rule: If an angle be expressed in grades, multiply the measure in grades by 9, divide the result by 10, and you obtain the measure of the angle in degrees.

Ex. How many degrees are contained in the angle 42o. 34`. 56“ ? 42o. 34. 56" = 42.3456 grades

9

10 381.1104

degrees 38.11104

60

minutes 6.66240

60

seconds 39.74400

the angle contains 38°. 6'. 39"-744,

EXAMPLES.-VII.

Find the number of degrees, minutes, and seconds in the fol

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41. If the number of degrees in an angle be given, to find

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Hence we obtain the following rule:

If an angle be expressed in degrees, multiply the measure in degrees by π, divide the result by 180, and you obtain the circular measure of the angle.

Ex. Find the circular measure of 45o. 15′;

45°. 15′ = 45.25 degrees;

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(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.

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Hence we obtain the following rule:

If an angle be expressed in circular measure, multiply the measure by 180, divide the result by π, and you obtain the measure of the angle in degrees.

Ex. Express in degrees the angle whose circular measure

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Express in degrees, &c. the angles whose circular measures

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