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III. (1) Find all the ratios of the angles

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(2) Find sin A and cos A, and thence find all the ratios of the different angles A, where

1

(3)

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2. tan A=0; sec A= 200; tan A

Find sec and cosece, and thence deduce all the ratios of 0, where

1

1.

tan 002; cot 0 = 20; cos 0:

=

2'

2. cot 0.125; tan 0=43; sin 0=

3 4

CHAPTER IV.

TRIANGLES.

27

To shew the immediate use of these ratios it will be seen that it is with these measures of angles and not with the arithmetical measures that the sides of triangles are connected.

Let ABC be a right-angled triangle. We may denote the angles by A, B, C, and the sides opposite to them by a, b, c.

Here A, B, C represent the number of degrees, grades, or the circular measure of the angles, and a, b, c denote the number of feet, inches or any measure of the length of the sides. So that these letters, as is usual in Algebra, represent numbers.

AC is called the side b, being B opposite to the angle B.

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a

C

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28

In the same way we obtain all the values;

a = c sin A, or c cos B, and also = b tan A, or b cot B. b=c cos A, or c sin B, and also a cot A, or a tan B. c = a cosec A, or a sec B, and also =b sec A, or b cosec B. These are true for the case of a right-angled triangle.

These values of the sides should be written out from the figure alone, over and over again, so as to make the results and the method of obtaining them familiar. Also in any triangle

sin A sin B sin C

а

This may be easily proved for an acute-angled triangle.

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29

30

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These formulæ may be used to find values of parts of a triangle when others are given.

A figure should always be drawn and the values which are given should be written by the corresponding sides or angles of the triangle.

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II. In any triangle,

sin A3, sin B = '12, b = 3,

given

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B

When one ratio of an angle is given, we have seen that any of the others can be found from it. But this does not imply that only one sort of ratio is necessary and the rest are superfluous, but the richness of the forms found for the sides of triangles shews that all are necessary because of the large choice of forms which they give for practical use.

EXAMPLES ON TRIANGLES.

I. In a right-angled triangle where C is the right angle

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