Example 2. The three angles of a triangle are in arithmetical progression, and the measure of the least in grades is to that of the greatest in circular measure as 120 : „. Express each angle

in degrees.

Let the angles contain x −y, x, x+y right angles respectively; they are then in A.P.

Their sum is 3x right angles; and since they are the angles of a triangle, their sum is 2 right angles;

.. 3x=2,

.. x=3.

Again, the least angle contains (xy) × 100 grades, and the greatest angle contains(x+y) radians,

π

2

.. 100 (xy)=60 (x+y),

or, 40x=160y,

or, x=4y.

.. 4y=,

because x=

or, y=f.

.. x − y = z − }=}, x=3, x+y=}+&={.

Thus the angles contain,, and right angles respectively;

therefore the angles are 45°, 60°, 75o.

EXAMPLES. XI.

(In the following examples the answers will be given in terms of π.)

(1) The sum of the degrees and of the grades in a certain angle is 38; find its circular measure.

(2) The difference of two angles is 20%, and their sum is 48°; find them.

(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.

(5) The difference between two angles is

is 56 degrees; find the angles.

π

and their sum

9'

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

(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: 3π. 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=}D.

(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 radians in any angle, prove that G – D=

200

π

(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 12o 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 is 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 9o 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 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 5o: 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.

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

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).