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(3) One angle is double of a second, and the sum of their measures in degrees and in grades respectively is 140; express the angles in degrees.

(4) Two angles are in the ratio of 4 : 5, and the difference of their measures in grades and in degrees respectively is 2} ; find the angles in degrees.

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(5) The difference between two angles is

and their sum is 56 degrees; find the angles.

(6) If the three angles of a triangle are in arithmetical progression, show that the mean angle is 60°.

(7) The three angles of a triangle are in arithmetical progression, and the number of grades in the least is to the number of degrees in the mean as 5:6. Find the angles in degrees.

(8) The three angles of a triangle are in arithmetical progression, and the number of grades in the greatest is to the number of degrees in the sum of the other two as 10:11. Find the angles in degrees.

(9) The three angles of a triangle are in arithmetical progression, and the number of grades in the least is to the number of radians in the greatest as 200 : 31. Express the angles in grades.

(10) If D be the number of degrees and G the number of grades in any angle, prove that G-D=ID.

(11) If M be the number of English minutes and m the number of French minutes in any angle, prove that

2M -m=M. (12) If G, D and C be the number of grades, degrees and

200 radians in any angle, prove that G-D=

(13) If an angle be expressed in French minutes, show that it will be transferred to English minutes by multiplying by •54.

(14) Divide 33° 6' into two parts so that the number of English seconds in one part may be equal to the number of French seconds in the other part.

(15) Find the ratio of 9° 27' to 128 50'.

(16) Find the number of radians in an angle of n English minutes.

(17) Express in each of the three systems of angular measurement the angles

(i) of a regular hexagon,
(ii) of a regular octagon,

(iii) of a regular quindecagon. (18) Show that the number of degrees in an angle of a regular decagon to the number of grades in an angle of a regular pentagon in the ratio of 6 : 5.

(19) Show that the number of grades in an angle of a regular pentagon is equal to the number of degrees in an angle of a regular hexagon.

(20) Find in English minutes the difference between the angle of a regular polygon of 48 sides and two right angles.

(21) If we take for unit the angle between a side of a regular quindecagon and the next side produced, find the measures (i) of a right angle, (ii) of a radian.

(22) Find the unit when the sum of the measures of a degree and of a grade is 1.

(23) What is the unit when the sum of the measures of go and of 58 is ?

(24) If the measure of b grades is a, find the measure of c degrees.

(25) What is the unit when the sum of the measures of a grades and of b degrees is c?

(26) The number of grades in a certain angle exceeds the number of degrees in it by t of the number of degrees in a radian. If this angle be taken as unit, what is the measure of a right angle?

(27) The three numbers which express the three angles of a triangle are all equal, and the units of angle in each are respectively a degree, a grade and the sum of a degree and a grade ; express each of the angles in circular measure.

(28) The three angles of a triangle have the same measure when expressed in degrees, grades and radians respectively; find this measure.

(29) The measures of the angles of a triangle in degrees, grades and radians respectively are in the ratio of 1:10:100; find the number of radians in the smallest angle.

(30) The interior angles of an irregular polygon are in A. P. ; the least angle is 120°; and the common difference 5° : find the number of sides.

CHAPTER V.

THE TRIGONOMETRICAL RATIOS.

73. Let ROE be any angle (see the figure in Art. 83). In one of the lines containing the angle take any point P, and from P draw PM perpendicular to the other line OR.

Then, in the right-angled triangle OPM, formed from the angle ROE,

(i) the side MP, which is opposite the angle under consideration, is called the perpendicular;

(ii) the side OP, which is opposite the right angle, is called the hypotenuse;

(iii) the third side OM, which is adjacent to the right angle and to the angle under consideration, is called the base.

From these three,-perpendicular, hypotenuse, base,—we can form three different sets containing two each. The ratios or fractions formed from these sets, viz. perpendicular

base

perpendicular (i) (ü)

(iii) hypotenuse hypotenuse

base and the ratios formed by inverting each of them, viz. hypotenuse hypotenuse

base (iv)

(v) perpendicular

(vi)

perpendicular' will be found to be of great importance in treating of any angle ROE. Accordingly to each of these six ratios has been given a separate name (Art. 75).

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74. The student should observe carefully (i) that each ratio, such as

perpendicular

is a mere number;

hypotenuse (ii) that, as we shall prove in Art. 83, these ratios remain unchanged as long as the angle remains unchanged;

(iii) that if the angle be altered ever so slightly, there is a consequent alteration in the value of these ratios.

[For, let ROE, ROE' be two angles which are nearly equal;

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Let OP=OP'; then OM is not=OM, and therefore the ratios OM ОГ

and OP ОР"

are not equal ; also MP is not=M'P' and therefore

MP M'P' the ratios and are not equal.]

OP

ОР

(iv) that by giving names to these ratios we are enabled to apply the methods of Algebra to the Geometry of Euclid VI., just as in Chapter I. we applied the methods of Algebra to Euc. I. 47.

The student is recommended to pay careful attention to the following definitions. He should be able to write them out in the exact words in which they are printed.

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