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2. In like manner, if the leg BC, Fig. 51, be made radius; then the other leg AB will represent the tangent, and the hypothenuse AC the secant, of the arc BG, or angle C.

3. But if the hypothenuse be made radius; then each leg will represent the sine of its opposite angle; namely, the leg AB, Fig. 52, the sine of the arc AE or angle C, and the leg BC the sine of the arc CD, or angle A.

The angles and one side of a right-angled triangle, being given to find the other sides.

RULE.

Call any one of the sides radius, and write upon it the word radius; observe whether the other sides become sines, tangents or secants, and write these words on them accordingly. Call the word written upon each side the name of that side. Then,

As the name of the side given,

Is to the name of the side required;
So is the side given,
To the side required.*

* DEMONSTRATION. Let ABC, Fig. 53, be a right-angled triangle; then it is evident that BC is the tangent, and AC the secant, of the angle A, to the radius AB. Let AD represent the radius of the tables, and draw DE perpendicular to AD, meeting AC produced in E; then DE is the tangent, and AE the secant of the angle A, to the radius AD. But because of the similar triangles ADC, ABC, AD : DE :: AB : BC; that is, the tabular radius : tabular tangent ::AB: AC. Also AD: AE :: AB: AC; that is, the tabular radius: tabular secant :: AB: AC. These proportions correspond with the rule. When either

Two sides of a right-angled triangle being given, to find the angles and other side.

RULE.

Call any one of the given sides radius, and write on them as before. Then,

As the side made radius,

Is to the other given side;

So is radius,

To the name of that other side.*

After finding the angle, the other side is found as in the preceding rule.

EXAMPLES.

1. In a right angled triangle ABC, are given the base AB=208, and the angle A = 35° 16', to find the bypothenuse AC and perpendicular BC.

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2. In a right-angled triangle ABC, there are given the hypothenuse AC = 272, and the base AB = 232; required the angles A and C, and the perpendicular BC.

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3. In a right-angled triangle, are given the hypothenuse AC 36.57, and the angle A= 27° 46', to find the base AB, and perpendicular BC. Ans. Base AB = 32.36, and perpendicular BC = 17.04.

4. In a right-angled triangle, there are given, the perpendicular = 193.6, and the angle opposite the base 479 51'; required the hypothenuse and base. Ans. Hypothenuse = 288.5, and base = 213.9.

5. Required the angles and hypothenuse of a rightangled triangle, the base of which is 46.72, and perpendicular 57.9. Ans. Angle opposite the base 38° 54′, angle opposite the perpendicular 51° 6', and hypothenuse

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