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CHAPTER III

RIGHT TRIANGLES

28. In every triangle there are six elements or parts. These are the three sides and the three angles.

When three elements are given, one of which is a side, the other three elements can be determined.

The solution of a triangle is the process of determining the unknown parts from the given parts.

In the present chapter it will be shown how the trigonometric functions can be employed to solve the right triangle.

29. Applications of definitions of the trigonometric functions to the right triangle. Let ABC be any right triangle. Place the triangle in the first quadrant with the vertex of one acute angle coinciding with the origin, and one side, not the hypotenuse, coinciding with the X-axis, as in the figure.

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It is seen that the functions of any acute angle of a right triangle are expressed in terms of the side adjacent, the side opposite, and the hypotenuse, without reference to the coordinate axes.

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30. Trigonometric tables. In the preceding chapter the trigonometric functions of 30°, 45°, and 60° were calculated. By processes too complicated to introduce here, tables have been computed, giving the values of the trigonometric functions of acute angles. These tables generally contain two parts. In one part the values of the functions, called natural functions, are given; in the other part the logarithms of the trigonometric functions, called logarithmic functions, are given.

When an angle is given its trigonometric functions can be taken from the tables, and vice versa. The functions of known angles thus become known numbers and can be used in problems of computation.

Approximate values of the trigonometric functions can be obtained by graphical methods.

90°

PROBLEM. Measure the distance, abscissas and ordinates of the points a, b, c, j. From these measurements

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compute to two figures the sine, cosine, and tangent of 0°, 10°, 20°,..., 90°. By arranging the results in tabular form, a two place table is constructed.

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This table may be used to solve Examples 1 to 6, Art. 36. On account of its extreme simplicity the use of this table allows the attention to be focused upon the fundamental

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31. Formulas used in the solution of right triangles. A right triangle can always be solved when, in addition to the right angle y, two independent parts are given. The formulas usually employed are:

sin ẞ=b

=

α + B = 90° a2 + b2 = c2

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When the computations are made without logarithms, the formulas involving the cotangent, secant, and cosecant may sometimes be used advantageously.

32. Selection of formulas.

(a) If an angle and a side are given, it is always possible to find the unknown parts directly from the given parts without the use of the formula a2 + b2 = c2. To find the unknown side, select that formula which contains the given parts and the desired unknown side. The unknown angle can always be found from a + B = 90°.

(b) If two sides are given, the third side would naturally be found by the use of a2+ b2 = c2, but in practice it is generally preferable first to compute an angle by the use of a formula involving the given sides and an angle. To find the third side, select a formula containing one of the giver. sides, the angle already computed, and the required side.

33. Check formulas. In all computations it is necessary constantly to guard against numerical errors. However carefully the computations are made, errors may still occur and therefore computed parts should be checked by means of check formulas. Any formula which was not used in the solution of the triangle may be used for this purpose. For the right triangle the formula

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may conveniently be used, and in general it is sufficient.

34. Suggestions on solving a triangle. Make a careful free-hand construction of the required triangle, and write down an estimate of the values of the unknown parts. Large errors will be detected readily, without the use of check formulas, when the computed parts are compared with the estimates.

Before entering the tables, and before making any computations, select all the formulas to be used, solve the formulas for the required parts, and make an outline in which a place is provided for every number to be used in the computation. This will often lessen the actual work, for frequently several required numbers are found on the same page of the table.

The arrangement of the work is of considerable importance in every extended computation.

35. Illustrative examples.

1. In a right triangle, given b = 14, a = 35°, to find a, c, and B.

SOLUTION. Approximate construction.

Estimate a = 9, c = 17.

By natural functions

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a

b

a=35°

COS α=

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..C=

B = 90°

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Check

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By logarithms

a = b tan c

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log b log cos a log c

log b

log tan a

log a

a

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