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61. In general, any one of the six trigonometric ratios of an angle being given, the relations expressed by the foregoing formulæ will enable us to find the value of all the rest. These are termed the elementary formulæ, and are collected in the following

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62. To find the SINE and COSINE of the SUM of two angles by

means of their sines and cosines.

I

G

D

Let the two angles be COD, DOE. In the line OE take any point, E, draw EF perpendicular to OC, and ED perpendicular to O D. Draw D G perpendicular to EF, and DC perpendicular to OC. The triangles GED and COD have their sides perpendicular, hence they are similar (Geom., Prop. XXV. Bk. IV.), and the angles DEG and COD are equal. Let a = COD = DEG, and b— DO E.

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63. To find the SINE and COSINE of the DIFFERENCE of two

angles by means of their sines and cosines.

Let the two angles be FOD, D O E. In the line OE take any point, E, draw EF perpendicular to OF, and ED perpendicular to OD. Draw DG perpendicular to FE produced, and DC perpendicular to OF. The triangles GED and COD have their sides perpendicular, hence they are similar (Geom., Prop. XXV. Bk. IV.), and the angles DEG and COD are equal.

Let a COD = DE G, and b = = DOE.

D

C

F

G

E

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64. The four formulæ last established apply to arcs as well as angles, and may be considered the fundamental formula of subsequent analysis. They are brought together in the following

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SIGNS OF THE TRIGONOMETRIC FUNCTIONS.

65. If on any line, straight or curved, different distances be measured from a fixed point of origin, the distances which have contrary situations may by convention be introduced into our calculations, by affecting the quantities representing their magnitudes by contrary signs.

A'

Let O be a fixed point in any line, A A", and suppose we have to determine the situations of other points in this line with respect to 0. The position of any point in the line will be known if we know the distance of the point from 0, and also know on which side of O the point lies. Now it is found convenient to adopt the following convention: distances measured in one direction from

A"

A

O along the line will be denoted by positive quantities, and distances measured in the contrary direction from O will be denoted by negative quantities.

Thus, for example, suppose that distances measured from O towards the right are denoted by positive numbers, and let A be a point, the distance of which from O is denoted by 2 or +2; then if A" be a point situated just as far to the left of O as A is to the right, the distance of A" from O will be denoted by 2.

In like manner, if distances originating in A A", and taken along O A', or only parallel to O A', when measured upwards be denoted by positive quantities, on being measured downwards they will be denoted by negative quantities.

66. A similar convention may conveniently be adopted with respect to angles.

Let any line, O B, revolve from the position OA, in one direction round 0, forming the angle BO A, and let this angle be denoted by a positive quantity; then, if the line OB revolve,

A

A'

B

A

B'

A"

from the position OA, round O in the contrary direction, forming the angle BOA, this angle may be denoted by a negative quantity.

Thus, for example, if each of the angles AOB and A OB is two ninths of a right angle, and we denote the former by 20° or +20°, the latter may be denoted by — 20°.

The direction of the positive distances is quite indifferent; but, being once fixed, the negative distances must lie in the contrary direction.

67. The representation of angles as the measure of the revolution of a line, turning in a plane about one of its own points, leads to the consideration of angles, not only greater than two right angles, but of all possible magnitudes.

Thus, when the line OB, starting from the initial position OA, has passed A", or made more than half a revolution, we have described an angular magnitude

of more than 180°; and when it has

passed on to A, we have an angular magnitude of 360°. If it now contin

ues to revolve in the same direction till A it arrives again at B, we have an an

gular magnitude of 360°+20°380°,

and thus we may conceive of angles

A'

B

A

'B'

A"

of all magnitudes. In like manner negative angles of all magnitudes may be formed by the describing line OB revolving from O A, but in a contrary direction. 68. The algebraic signs of the trigonometric functions can be readily fixed in the mind by being represented geometrically. Thus,

Let the extremity of a revolving line, starting from the initial position O A, describe the positive arc AB less than 90°, A B' between 90° and 180°, A A'B' between 180° and 270°, and A A'A"B" between 270° and 360°. Then, according to the definitions of Art. 54, BD, B'D',

Α'

B'

B

A

D

0

D

B

A"

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