The circular measure of two right angles is π; the circular measure

π

of one right angle is and the circular measure of n right angles

2'

where n may be either integral or fractional.

22. We will now shew, how to connect the circular measure of any angle with the measure of the same angle in degrees. Let x denote the number of degrees in any given angle, the circular measure of the same angle. Since there are 180 degrees in two

x

right angles, expresses the ratio of the given angle to two right

180

Ө

angles. And since π is the circular measure of two right angles, also expresses the ratio of the given angle to two right angles. Hence

23. For example, the circular measure of an angle of 1 degree

π

is the circular measure of an angle of 10 degrees is 180

circular measure of an angle of half a degree is

cular measure of an angle of one minute is

measure of an angle of one second is

π

180 × 60 × 60

and so on.

degrees contained in the angle is

Again; if the circular measure of an angle is

if the circular measure of an angle is 10, the number of degrees

3

contained in the angle is 10.

180

П

that is 10 x 57.2957795...; and

so on.

The student is recommended to pay particular attention to these points; especially he should accustom himself to express readily in circular measure an angle which is given in degrees.

24. Similarly we may connect the circular measure of any angle with the measure of the same angle in grades.

Let y denote the number of grades in any given angle, the circular measure of the same angle; then the ratio of the given Ꮎ angle to two right angles is expressed by and also by

Hence

У 200

π

The number of grades in the angle which is the unit of circular

where nothing is assumed respecting the unit of angular measurement, except that the same unit is to be employed for both angles. Since AOB is an invariable angle, we see that the magnitude of any angle AOC varies as the subtending arc directly, and as the radius inversely. Thus we may say that

when k is some quantity which does not change with AOC, and the value of which depends upon the unit of angular measurement

which we please to employ. Suppose, for example, that we wish to take the half of a right angle as our unit; then we require that AOC should be equal to 1 when the arc is the eighth part of the circumference; thus

gives the correct estimate of the magnitude of an angle when the unit is half a right angle.

1. If D, G, C be respectively the number of degrees, grades, and units of circular measure in an angle, shew that

2. Find the number of degrees in the angle subtended at the centre of a circle whose radius is 10 feet by an arc whose length is 9 inches.

3. Find the circular measure of 1o. N.

4. There are three angles; the circular measure of the first

π

exceeds that of the second by the sum of the second and third

10'

is 30 grades, and the sum of the first and second is 36 degrees. Determine the three angles.

5. Express five-sixteenths of a right angle in circular measure,

in degrees and decimals of a degree, and in grades and decimals of a grade.

6. The angles of a triangle are in arithmetical progression, and the greatest is double the least; express the angles in degrees, grades, and circular measure.

7. The angles of a triangle are in arithmetical progression, and the number of degrees in the least is to the circular measure of the greatest as 60 to π; find the angles.

L

III. TRIGONOMETRICAL RATIOS.

26. Let BAC be any angle; take any point in either of the containing sides, and from it draw a line perpendicular to the other

side; let P be the point in the side AC and PM perpendicular to AB. We shall use the letter A to denote the angle BAC. Then

If the cosine of A be subtracted from unity, the remainder is called the versed sine of A. If the sine of A be subtracted from unity, the remainder is called the coversed sine of A; the latter term however is rarely used in practice.

27. The words sine, cosine, &c. are usually abbreviated in writing and printing; thus the above definitions may be expressed as follows,

28. The sine, cosine, tangent, cotangent, secant, cosecant, versed sine, and coversed sine are called trigonometrical ratios or trigonometrical functions; sometimes they have been called goniometrical