2. Find the 7th power of 8. Ans. 2097154, nearly.

EXTRACTING ROOTS BY MEANS OF LOGARITHMS.

19. From the principle proved in Art. 8, we have the following

RULE.

Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required.

EXAMPLES.

1. Find the cube root of 4096.

The logarithm of 4096 is 3.612360, and one-third of this is 1.204120. The corresponding number is 16, which is the root sought.

When the characteristic is negative and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa.

2. Find the 4th root of .00000081.

The logarithm of .00000081 is 7.908485, which is equal to 8 +1.908485, and one-fourth of this is 2.477121.

The number corresponding to this logarithm is .03: hence, .03 is the root required.

PLANE TRIGONOMETRY.

20.

PLANE TRIGONOMETRY is that branch of Mathematics which treats of the solution of plane triangles.

In every plane triangle there are six parts: three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts, is called the solution of the triangle.

21. A plane angle is measured by the arc of a circle mcluded between its sides, the centre of the circle being at the vertex, and its radius being equal to 1.

Thus, if the vertex A be taken

as a centre, and the radius AB be equal to 1, the intercepted arc BC will measure the angle A (B. III., P. XVII., S.).

A4

B

Let ABCD represent a circle whose radius is equal to

1, and AC, BD, two diameters perpendicular to each other. These diameters divide the circumference into four equal parts, called quadrants; and because each of the angles at the centre is a right angle, it follows that a right angle is measured by a quad

D

rant. An acute angle is measured by an arc less than a quadrant, and an obtuse angle, by an arc greater than a quadrant.

22. In Geometry, the unit of angular measure is a right angle; so in Trigonometry, the primary unit is a quadrant which is the measure of a right angle.

For convenience, the quadrant is divided into 90 equal parts, each of which is called a degree; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are denoted by the symbols °, "". Thus, the expression 7° 22' 33", is read, 7 degrees, 22 minutes, and 33 seconds. Fractional parts of a second are expressed decimally.

A quadrant contains 324,000 seconds, and an arc of 7° 22' 33" contains 26553 seconds; hence, the angle measured by the latter arc, is the 55th part of a right angle. In like manner, any angle may be expressed in terms of a right angle.

23. The complement of an arc is the difference between that arc and 90°. The complement

of an angle is the difference between that angle and a right angle.

Thus, EB is the complement of AE, and FB is the complement of AF. In like manner, EOB is the complement of AOE, and FOB is the complement of AOF.

In a right-angled triangle, the

F

acute angles are complements of each other.

B

A

24. The supplement of an arc is the difference between

that arc and 180°.

ference between that

The supplement of an angle is the difangle and two right angles.

Thus, EC is the supplement of AE, and FC the

supplement of AF

ment of AOE, and

In like manner, EOC is the suppleFOC the supplement of AOF In any plane triangle, either angle is the supplement of the sum of the other two.

25. Instead of employing the arcs themselves, we usually employ certain functions of the arcs, as explained below. A function of a quantity is something which depends upon that quantity for its value.

The following functions are the only ones needed for solving triangles :

26. The sine of an arc is the distance of one extremity of the arc from the diameter, through the other extremity. Thus, PM is the sine of

AM, and P'M' is the sine of
AM'.

T"

T'

If AM is equal to M'C, AM and AM' will be supplements of each other ; and because MM' is parallel to AC,

M'

N

MT

P'

PM will be equal to P'M' (B. I., P. XXIII.): hence, the sine of an arc is equal to the sine of its supplement.

27. The cosine of an arc is the sine of the complement of the arc.

Thus, NM is the cosine of AM, and NM' is the cosine of AM'. These lines are respectively equal to OP and OP'.

It is evident, from the equal triangles of the figure, that the cosine of an arc is equal to the cosine of its supplement.

28. The tangent of an arc is the perpendicular to the radius at one extremity of the arc, limited by the prolongation of the diameter through the other extremity.

Thus, AT is the tangent of the arc AM, and AT"" is the tangent of the arc AM'.

T"

B

T

M'

N

MT

If AM is equal to M'C, AM and AM' will be supplements of each other. But AM"" and AM' are also supplements of each other: hence, the arc AM is equal to the arc AM", and the corresponding angles, AOM and AOM"", are also equal.

A

The right-angled tri

angles AOT and AOT"", have a common base 40, and the angles at the base equal; consequently, the remaining parts are respectively equal: hence, AT is equal to AT"". But AT is the tangent of AM, and AT""" is the tangent of AM' hence, the tangent of an arc is equal to the tan gent of its supplement.

It is to be observed that no account is taken of the alge braic signs of the cosines and tangents, the numerical values alone being referred to.

29

plement.

The cotangent of an arc is the tangent of its com

Thus, BT" is the cotangent of the arc AM, and BT" is the cotangent of the are AM'.

The sine, cosine, tangent, and cotangent of an arc, are, for convenience, written sin a, cos a, tan ɑ, and cot a